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# MCAT StudyGuide - Physics

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UCLA COURSE READER PRINT DIGITAL MULTIMEDIA Mathematics 135 Differential Equations 2nd with Applications and History NO REFUNDS NO EXCHANGES DIFFERENTIAL EQUATMNS WITH APVAPLICATIQNS AND HISTRICAL NDquot ES R E PHI NTEUlI W IT H PERM I35 I UN 1 F Pgzi FvIuGraws39Ili1 H 9 En HquotE Tax Q r Di ffEI a3111ial Eq uati CIETIS wi th Am in Iicn1a U U Sm1nm39I15 U39ED39gE F 3 j IEJEM MnGrawHIil lnternatio nal Series in Pure and Applie y A hlIms s rjmpfex Anuquot395IL5 Em39jier and Aduaaged M Ih m i39EH5 f Ihmi fimvr 5Ef 7 arltd E39ng nEer5 Buas mvHcrnn In CIomp er Anafy5 s B39u1cI zquot1d39lquot 39H39EEdiCEEEEEEHE llZ u llnn Parim D i EF39E39 If f EquaIm4m lE nllE and dE oIjImr EfEmEnrarr NuKmErfcm39 AnuIyfs Ari Ai39guraltthrrIi E Apprmigtrh E Eli 3I Elfi iKE 39EIi aiFham39 7m M ad 1 s in B 39a gjy Goldberg Matrix Theary wm1 lppfquotetar39mn3 EIp39 FfPH Nf in CmpuraIi4armf Matrix AJ39gebra j gwin and An InFr dMfI n rm ampM39arhemaH7mI An a39 ysf i iMmum39a3l1 Brii g rm bstfrmif M39a hEmaiIinc sfkii and Etipiez In roduLcr nV In M39arhemarimu39f Ana y3Ii 1Equotiljskr Pmr39Ic m39 D er39ent39 a39 E quar 3r1rm anus amdur1y Vain P b39 fT mm Appiirltmi Lani5 Fiinlerz Bank rinf bsrfr cr 139gEbra R ElEVl Ind R hi39lI Hquoti39I Ii A Ffm CQHFEE in Num1 H39m39139 A nufy Lf Hilgfer and R nsee D gr4 nt7 EqM i 5 Hmh AppJimI inrL g Prin fpfes f Marh ma r cmF Ana y5is R udi RE1 39 and C39 mpF r Anm 539i39r Simmnnsa D39 39TEre4nriaJ Euqfu3I n2mr withApp39irarmr15u39nd H39 5rarEcu Nam Smaa vmcl Husnck Cm rzuim An IntraredA4ppnm1 Vnnd e4n Ejmdan EaEmEnm3 Thwrjw Wi5 er Imrmiu cnl n InV Ab3 racr 4 1cJ39g bltm CIl1urchilVf Bmwn Series mpu 39 x V rinbJEs5 E 39T ad App HE39 iti FM F un39Er 5Eriei mind Buundary Vafua Pr mnmi p rarimna M r hemurEL Also Available from McGrawHill Schaum s Outline Series in Mathematics amp Statistics Most outlines include basic theory de nitions and hundreds of solved problems and supplementary problems with answers Titles on the Current List Include Advanced Calculus Advanced Mathematics Analytic Geometry Boolean Algebra Calculus 3d edition Complex Variables Descriptive Geometry Differential Equations Differential Geometry Discrete Mathematics Essential Computer Mathematics Calculus of Finite Differences amp Difference Equations Fourier Analysis General Topology Group Theory Laplace Transforms Linear Algebra 2d edition Mathematical Handbook of Formulas amp Tables Matrices Matrix Operations Modern Algebra Modern Introductory Differential Equations Numerical Analysis 2d edition Partial Differential Equations Probability Probability amp Statistics Projective Geometry Real Variables Set Theory amp Related Topics Statistics 2d edition Tensor Calculus Vector Analysis Available at your College Bookstore A complete list of Schaum titles may be obtained by writing to Schaum Division McGrawHill Inc Princeton Road S1 Hightstown NJ 08520 The miss aim of 3c Ence 5 rim hnar of the human mmd and fmm Ihti pain Hf view a queixiion abcrur rmmbers 2 2 imp0r39 mI as H qViue3 ion afb ui the System f the w irl i J Jambki p p p HITORICAL NT p Second p Genrige Prafa 3 r uztirf Ma he matic5 Cm39mr dlcr C u9Hege X 3 new chapmr on numerirsal I39l IEIihDdS by John P Robertson D39Ep FfTIE f MarhEma Ecm39 Sif39f HEE U ni Ed Srares Mimary Academy McGrawHill inc Nrw 1rk St Lsuuisx San39Fram391ti5um u u39lIanL fn BDgzwt Carmzasl Hamburg Li5hrrm Ln39nndmnn Madrid Mefxinzgacr M Man E Io39n1139E3 New Balm P39E1iS San Juarn Sa 1urPa LmlJ Singap rre Sydney Tc1ljr1 Titimjnm J P wa5 SEES In Tim 5i R manw diir39mr5 were 39Riehar39d Wa is and Iuhn Mmri5539 ihE pruduc uEum stugpcarvizmr was Lani5 Earam wnwe39r39 d ieii Z ji by Ema Bauaeurq iF39rmjecI sugperviisinzim W35 d by 111 Univcr5iiIics Fmss P DnnLnlh3r 5 Suns A umpany W325 pninler and bindarr l2llIF F E QlJATLl NTS LPPL C L T iDlN5 HII5 TfRICampL NGTES Cu mrr39ighI quotWIFE by McGrawH39iiL InE All rights 39mscr39vgdl Frinteuda in the 39LEnit cd Sitalcs urnf mquot ur riIEltr39it39I quotEmEpI as pcr mitted und r rlhe Limited EsiImtamp Cfupy rigfht M21 I jf no jpan uf H1Ii pubi min n may be rc1prroduEmi cur di5ampjributevd in any fusrrm Dr by Emy mans BF s39IurrnIl in 1 data base or retrmrtail rstmquot wtthnuI H1eE pzmjr writmn prmi55i1n Lil IITE1 puhWis11erg quotll 4 5 tr 393 S W I39 DDC P 395 4 2 11 U IEN D EEquot4D l Libraampr3r f C igfl ifi EaIalngimg inePuhliAcaliimm alia Eiimmsn5 iiixecgsm Fin13Jjh nIdaIej Dif2JfirerLtHIsl EquHI Iinnn5 w39iEh arpIi 39a39li nss and histt1nlaafn iJ I1I 73939I M39 E tt39Lwrgu F SJummL1 mmv 2n EI Ed CFTIJA lWlT5 5 iI 391 1 TlJiafE39renIiial EqLiiEiitJ39n2i II 39I Me MY 5IE9 quotIquotWI 5ehE3935m JV2 mi33rHa ABOUT THE UTH R Geurge Sil liilllrll fhas aciadeimjiiat degrees frvam the Caiifmimia In5tiiituite of iTEirhinci0g3r P Uiniweriaivty of i hiicagn and Yale Univer siitL Hie tiaughit at several wliiieges and uinivesiriisities befurei joining ithe facumr of Coloradin i oliiege in 1962 WihEilquot39E he is Pmfessimr inf M thETm 39iCS He is aim the lItiquotl Iquot If LI irducIi0ini Err T pafay and MadErn Anm39ya39E3 MEGmw iHLill 1963 Pr 39C iC ifu Ma39t1emaI39ic5i in a Nu sihe f Jan5 n Pubiicatianns 1 981 and C lvCHfH3 with Anm y I39ui CFEmmeIry MvGraiwiHilL p When not iworkiing I talking or eating inr driniltiing or czmrkiinig Pr fessnr Simrmns is likely to be itiravelinig Western and Saiutheirn EumApe Tiurkey Israel Egypt Russia China Southeast Asia tmut ahing RiDciiky Miauntaini stateaj 39pquotiayiing pocfket biiiiiarsi Ur reaiing liiteiraitum liiisitmrgw hi griaphy d auitabiiograhy Ecience and enuugh iihriiieras tr ac hieve u ji jv mE t without guifliti Prefiee to the Scend Editien Preface to the First Eiditiien Suggestions for the Instructm The Niaitiupe ef Di ereintiiial EC11 ti3I1S Serpareblei iEquatims 1 2 3i 4 5 is I39ntreduetien General Remarks en Selutien5 Families ef Curquot39ir39eei Orthegnel TI39 39jiE ED EEES Grew39Ih 39ece3r Cheimieei Ree etit nei and Mixing Falliing Becliies and Other Metien Pzreblefms The Etreehietesehrene Fermet and the erneu ie First Umere Equartimle quotF P ID 11 Hemegeneems Equerliem5 Exact Equelier1e Ii1 teg1ating Fae Ie1 s Linear Equetiees Reduetieii eff Order The Hangii1g Chain Pursuit iCu rvee Simple EIieet139ie Ciireui I s Second Order Lixnear Equatielis 14 Ii 16 W 13 II11iI dUEEiU39 The General Selutiien ef the Hemegeneeue Equatien The Use ef a Knewn Seiutiein in Find Anemer The 39HemDgeneeue Equetiien with Ceinstem Cee eiente The Metheind em Undetermined Ceef eient3 EW xvii mi 10 1739 29 35 4 4 51 54 63 022 T 1 N1 8 92 95 99 Ill The Mctlmd of Variati on Elf ParamE1ters Vib ratians in MEKchaniuaTlV and E ect nml Sy5Icms EL Newmaunquot5 Law at Gravitamii nn and the Mmim1 of the PLane1Is PA quotI I39igher Drrader LiLnrEa1r ua t1iuIi5 LCun u pIad I iam mnnriu Gssuzillamnlrs 23 Op eratmT Mctzhndis Lima F indIirmg F ar39Ii ul ar Solutions Appenilz t Eru r Appendix Newmn Quali m1tAiTve Propertias Uf S0lutions OsuilampIatiuris and the P rm Sepamtimcn Thmrcm 225 7 Stuml Cam pan snn Theuirem PD WBI SE 1li65 SA l1Mians and Special Functimms K IAmtr udsutirmA F R ViEW of PGWEI Series 0 SEIiE5S 1L39lti 5Df F ir 5l Urdcr EquaIiun5 0 Sergznnd DrdEr Lineaar Equti0n5 Jrdina r3r Pnian ls Re lar39 Sin1gular Paints Rcg1u1arSin grLnlwar F39ain4ts C ntimucd 311 Gaussquot5 Hy39pergeamctri E39qIlHt5i39U N P J T1he4P ine1 at Mn niw Apjp n ix A Tum C nn verjgenCE Pf39 39f577 Appendix B Hermite Pul3rnU39inias and Quxantum MechaniE5 Gauss 39CThEbShEv 1PnEy nnmi4als arm the fMinsimax Prnpcxfrw Ei smann s Equa39t39i n AppEndi C 4 p Endix D Appendin p F 011rier Seriues and rth0 g0nal F uncti 1ns 33 The fFmmri er 39ff E ricnt5 P 3 THE Pr hham Elf C5 nsn wrge rm 35 E r39E Tl and Udcd FuncEVim1s Cusine and Sin Series 365 EJ IEII395i D ED AI7bii 1quotary I39m39I rval s 3 Orth0guna FungctinnA5 pit The Mlt Cmnverge nc e nf Fm urier Series Appm1Adix X7 0 t fTFixntwisE C i1 n39 Fg 39C1E Theorem Partial Vifferentiai EuatiVDnss anwil Bwmlndary Value ProVblemg 3 IL IriItI3913Id lIISIii iEIIFL Irist1ia45 REmm39k5 amp1 EignvaluE5 Eigven39functin39ns and NthE Vih393 Ili g Sv39lrri39ng 41 Heat Eq1mammn 43 The Eiru111leI VPar1v1lemV fur a 39 I Ea P i39 L n L5 Intc4gra S t uHrn1 LiouviIfle Frnblcms 03 W6 Ill 1115 11122 12 136 146 135 155 161 mss ms N2 M5 M34 192 J J 2 e e e fl 0tP e 246 E5 2792 2 0tP E93 391 I 3 1 T 323 11 P8 Appendix The Eimtenvee f EiElquotIquotia39 l1L39IES end EigenfunetiU39ne Some Speeielt Fun cti 0ns 0f Meeatfhemeatieeal Physics 44 45 45 41 iLegendre iPelggnemeiaEee hPrr per39itie5 U f Leegaendere P nlyn mial5 E e5eel Funetiene Gamrne Funetieen Prjperties ef Aeesel fumetiezmsj Appen dix f Legeendre Ptul39jrtrmrnieI5 c g d Perte ntiel Theery ApepverIdi x E Eteeeele F uneVtieene and the Veihre ti eng Meemmanwe Appenedie D Ad diiienal Preperties of Bessel Fui1eticrn9 Lapleee Traneferems 43 k Mntr dueti n Few Remrks en the Thenry Apjp lfieeeteiens to Di eueen tiele Equetieins Derivreetivee and Inetegral5 emf Laxplaees Tmnetfierme C enve leutiene and Aebel e Me henei eeel Premem Mare about Cmrwiemire ne a The Unit Step and rnpu1ee F urmtine Appeenedie A Leeepleee 39ppendix E Abel Sryslzems f First Order Equeaiien5 55 ST Geneeere Remarke Urn Syet e me LLinear Syrsteme Hemegeneeeeue Linear Systems with Coneteeni Ceee ieients Mnnlineaur Syeteeme VeE39terra s PreyPredeeeteeere Elquetiene ieevonlieneer Equatimis 58 59 61 63 Autenermeus Syeteems The Phase Plane and Its Pheeenem eene Types emf Ciritmele Peeinte St fbeihtjf Cri tiee Paints and StehiIi1yfer Liznear Syeeteme Steebi itey by Liepeunev ee Direert Meethed Simpe CriIieaT Paints nf Nenflineer Syete NrJnIineer Meeehainies Ceenseeervetivee Syeteme Per39iedie Sl lmlt i The PUineeer BendieSGn Tiheeverem App endix 8l Perinmr Aepvpieendrie P 1391Jnf39u339f Lii ne refs quotlF39heerem The Calculeues of Variati0nes 65 m Intrduetien Suane Typieceil Pmbleme g the Subject Euler e f i ereentie Equvatii3I1 fetr an Eetrermal 331 335 pgc 342 343 358 P j 3 1 31 381 331 335 3939394 33999 p 412 413 41139 421 4392 434 44 440 446 455 465 am 430 436 494 A49 502 502 505 cY PAX 6 I5 39p E139imEtrEE pmblams ppendix A Lagrange 524 Apfp ndilli 1 H amil39tm1 s P rVincipWc and Its Impl irHti n5 526 quotThe Exim nce and Uniq39u en e5s of St1uti2 ns 533 gY Thaw Mcth d of Sucue55i139e Appruximatians 533 X P iEard s ThEn39rrm 5413 quotA TE S3sI Ems Sec nd U rdr39I Linear KEquatinn 552 14 NumEIicaI Methods 5545 I l i Intrrwdusr1t icn 55 TF2 The M hnd of EUIEI T 73 Ermrs Ayn ImprmFEmEni in EW391JEIquot 0 O T5 H39 ig11ErOrder MEtI md5 S69 Sr5t em5 5 S Numerical Tables ST AnAswve1is 55 Ind ex 61 w ptf THE SECOND EDITIN Pof eerreet es e seeend ediIiri se gees the idiern I eerseinilggr hepe se ends I else ihepe thet enyene whe detects an error wilil de me the kindness ef ieittirng me kne we see that hrepeeirs earn he needed As Ceenfuieius said A man who mskesi er II1iSta3i B and deiesrft eerreer it is making tiwe misteiies I new sundieristsnd why seeend editiensi ef etie rtbeeiks ere always lner than first editiens as with gevernrnent39s end their budgets there is eiwsiysi streng pressure from lehhiyists to put things in bet rereiyr pressure In take things emit The resin ehenpes W this new editiien are as i eIews the number ef prnblerns in the rsit pert ef the beek has been rnere i hen de nhled there are twee new ehepters en Furrierr i5Series end en Pertiei Di 39erenitiei Equetiinnsi seetiens en higghieer erider iineer equeiiens end epeireter rnetheds have heen edded te Chapter 3 end sfdurthier rneteriel en CD V 39iL1iZi S end ernineering eppiieetiens has been added re the chapter en Liepleeie Trensferrns AEtegetheer meen different Uvne sxemestxer eeerisesi can be bsniir en esriens perish ef A heeic by using thee sehen1etie emieirne ef the ehepters giiveen en page ssL There is even seenugh rneteeiriei here for e itwei sernester wuirse if the app endiees ere iieken inte aeemnnt F inieilnri an entireliy new ehepter en Nnrnerieel Metheds Chapter 714 has been wrhtene espeieieliy for this editien by Mejer Jehn pN2 Rreheaertsenr ef the United Stetres iMiliter39y Aeeiden1ye Mejer Rehertsen s expertisie in these matters is rnueh greater then my ewfri and I em sure that imeieiny users ef this new editien will epspreeiette his eentriibutiein es I de MeGrsw iHiiii and i wenled like it ihenk the feiieeing reviewers fer their many heei39pfurl eemmeents end suggestsienisi Arterihursn3 Neiw W 3f1quot39i mi P rtEFa E T D THE SEEDHD Enmmr Maxim 39Tech LE dward acken5t eiJm I11n395 Univer5ity Hamldi Carda Smith aknt1a SEMJDAUI nf M in ES anti Techn 0l Dgy Weminng Chen U niver4sJity of Ariznmax Jerald 0 DauM Uniivcrsity M TAeVnnVesszBe Lssiteir B FuHer Rnchester Insltimte of Techml gy Juan A atic Unive1395it4y Elf I wa Richard 09 Herman The PE Si39F WV Li State UnivEr my Rc1gEr PA Marty C evelantd Ema UmivEIsity39 1FeansTPiBIre Meyer J hns H pkins Umiversirm Krzy szwf O5t 5zEw5ki UnAiwrsViAty inf LuuisviMe4 J pS Rmrnyak Univer5ity Qf Virjginia Man 4S harplE5 News MeAxim Tech BBIquot fEIi7El S11i 39man The Johns Hnpkins UnivEArsity amx CaLyin Wi cm Umversity mi G7eaQrg A i mmnns PREFACE TO TE Te be wserthy ef sertifes sttenitireen is new testeseke en en eid seijeet sheuid emtbed3r39 res drei nite and ressensblte peint ef view iwrhiieh is net represented E heelis iiIquotEHCiquot in print Such a pertinent ref View inevitsibiy reflects the esperientee tests and biases ef the iLJth iITi see si1evuld theretere be eieerly stated st the beginning se the these whe disagree can seek neurishrnent elsewherer c s39truuture sane teentents ef this heel expresis my persensl epinitensr in as vsrriretfy ef wsysrr es telews The piece es di eremial eq ueli ens u mathemelie151 Ansritysirs has been the derninent brsneh ef n1athernsties fer 30U y eers send dsifEfferrenitisl equetiiens are the heart ef snelysisr This s ubjteet is the I1 t r i gee ef eleimrerntery eeieulrest and the rarest irnpetritent pert ef metrhrenisties fer endetrsettsntditeng the phlysieefl seiesnees Aiset in the deeper questions it generetesr is the sevuree ef rnes t f the ideas and theeries wiiriseit eeestiitete higher snrariysissrr Fewer series Eeerier series the eaznnis fenetiteyn and ether stpeeisi funetir ens integrsfl e qerstiens existence theteriernsr the need fer rigerees justi estsierns ref many ansiytire preeeesstes eil therset Ii39lEviTif1EltS srise in enr WDIIIE in therir meet nsturei eentest And set E later tstegget ttley preside the fpriineipsl mretrissetien behind ee rnpir ers srsnsi3tsiis the stheery ref Feeriegr series end mere general etrmegrensri erpansieinsi Lettierssgee int egrstien rnetrie spaces and Hsiilberit sstpsees st iiIEJSWI ef ether ieeeu39tifel tepies in rneciern rnrsrtriiemrstieys iwiewuld srgee fer essmpl e the39I ene ef the msiin ireess ef eernpiezsr rsnrsiytsis is the liberetien ef perwer series freer the een ningr envitenment et the rest neniber systeLm end this nrrtetivre rnest teitetsrly felt R these whee iisave tried re use reel Z39I39 quotWET39 series te Si iltvi di ietretntiiel eqtus tiie nsi in betsnryi 0 is evieus ti Eli nee ene ease ieillry sppreetiste the Iwlessemst ref ewsstring pisnts witheet s reresetnsrbie 1i1ndeeristitseding ef the rieets stems end ieeses whsieiht nieurish ends sepptert th E me The seene prineipile is true in rnsr tfh en1tetiesr 131 is eiftien negleeted er fergttten 39I J quotirquot39ii tt tii i PREFACE TE THE FIRST iEiZl39iTID139HT Feds are he common in miatthemetttises es in any ether htuman etetivityh am it is eiweye di ieeuit tU Separate the enduring fmrtt the ephe39meral in the iChi tquot I I1EiitS hf erieie ewe tithe At presettt there is e stteiig eurrent hf ahstreetien ewihg threugh em gredeutattte sehehls ef tnethemetiee This ctitreht hate emuretlt away ttten3r ef the intiivitiuel feetmtee ef the iE139I39id5 EHpE an ri repteeetd them with the emeetti mutntded ihieuildtere eat gerierei th B riE is When taken in tI 1Dd Iquot Ii I1 these geetietel theei39iee are hethi ueefui and eettiiefyitngg butt DEE tmthttunete e eet emf their pres tl ilitil ie that if a student de eeh t ieerh e iittie white he is an il39lii 1d l39 gI tICi1lEJ E aheu39t eueh eeihttul and 39wetthw hiie ttI1pit39 as the wave etquetihh Gause e hyp39EIgE iltEIi Fi39E f11I39l39C3tiCltII the ga I I39IIII1E fuit1e tien and the basic prebieltis et the eeittuiius tit verietiens et meng many e39thets theni he is unIiiietiy the tie 5039 letter The rteturei piece teat an informal aequeinttanteet with 51ieh ideas is a leisurely i39I1 1397I DdLit3it71ii39jr E tlr mi ditfertetnttielt eqeetiehe Seine ef etit eurt etit heeite on this euhjeet remind me hi 3 eighiteeeing hue twhnee driver is 50 mhteeesed quotwith epeetling elnng the meet a S h t itli tihet hie paeeetngere heave little erquot rm eppesrtunity te enjtzry the Seenet3F Let LIE be late eeeeeiitjihally and take gir i iquot pleasure in the j itf 6 kppliEMi IS It is e I1quot11iSTt1 that n thitig is pejrmehettt eiteept ehes nget and the primaty li1 ptI1lSE ef ii etehtiel quetttietne ie ta Serve as a tee fer the study hf ehetige in the phyerieel werid A g t1E39t39 i heeit mi the euhjieet w itheut e teasetna39hie aeeeunt aft its eeieriti e 4ppii39C TIiQEnS wmiltl thtet efm39ie he as futile and peiritieee as e tt39eeti5e en eggs that did tiet m enttien their i39epredutttive purpeee Thie beak ie tetmetmtetet 50 that each ehepter BJ39IEEE pII the last has at least nine H39I jU39If peyh tfquot end erfteh eevereit iin the ii ftn set a eleeeic Ei iCiIE l39lTiii1ii39quot1t3 prcrhiertt w hith the ttteeihcrde of ethat Chapter ti39Egti39idEI actuzeeeiblret These pp39iit3 ti quotI1S ittCitilfiiB The bl39 CiT1iStOChIquotUi E prethlem The Eitrietetiin fetmuile E meg Itiewteh ts law hf gretvitetie h The wave eqmetittn fer the vihrieting string The hairrrmhie eseiitletert in quantum meieheniee PIfJquottEtTIIiai theory The wave equetteien fer the Viib39J E1 iii391g memhtet1e The pttety p re tjettnt eqjtietiehe Aquot Nonlinear nteehehiiee HamiiteIt391 e ptquotiit39tIEipIEv Aheti e meehenieeli prehiem I eensiider the metihematiieal MquotB I1T1 Btii et tiheeet prehilerhe ED thee emeng the chief glteries hf Wv IEPH eitriiiea39tien and I hepe the i EtH 1ElT wiii agree PREFACE Te THE Fisst e eiT1eel Yb The prnhlem sf tl fl lII lEll a39liI39HI tiger On the heights ef pure rriathtemtatiies any argument that purports tn he a jpreef nrmst he etap ahie et withstiandiing the 5iEquoti39El39ES lZi eriticisIns vet skteptiieiali experts This is time nt the suites eat the game and if yes wiisthi I play yeti must ahitde hy the miss But this is net the nnliy game in ttewn There are same parts of matihemiaties petrhaips nusnher theses and ahstttaet algebriae in which high stanards ef rigemns prentf amiss be aptepritate at alt levels But in eilenientaiy di f39ereniitiail equatiienits at narrnw insistentee nn dnetrinaire EKHCiiilI39i39ld tends te squeeze the juice tent ef the stihjectt sn that nnly the dry hush ternaizns main purpose in this beak is te help the stiudent grasp the nature and signi eaneet ef i erential equatiens and be this end I much prefers heing ecieasinnaiify itnipreeise but unelerstandahile he being tC tflJfiBI39EI3 accurate hint ineenfs ptehensihiet in am net at all interested in huiidding ta ingieaiiiy tinripeeeiahilet rnathenztatieal structure in which de xniitiens tiquot JE4 l39E Ii39I39iiS hand iigere t1s ptnefs ate weided tegether into at ifern1idahiet harrier whiieh the reader is ehaliened tee penettate In spite ef these diseiaimersi II de atternpt a fairiy I ig39DfUUS disenssinn fmm time tn time nntahly in Chapter 13 and Appendieest 0 in Chaptteirs 3 6 and T and in Ci IaptEI I are net saying that the test sf this heels is nenrigereus but enijy that it leans inward the Ett339iZi v39i5t seheuni nf n1athematies whese psimtairgit aim is ten detreiep inetheds fer snisinig iseitenti e iprehien1s tin eentrast tn the eetnttempiatiive sehierelt whiieh anahraes and iganiaets the ideas and tents gtenerateci by the iaetiisistis Seine will that a ntatheniatiieai 139gl1lII7139E i either is a preset er is nets a preef in the eentext of elementary anialysis I disagree and heilietre instead that the prepner reie ef st pteef is tea chatty reasenahle eentriieititen t eneis intended audience It seems te met that rnath en1atieaiIi rignir is igiiiiE elethiing in its sttyie it naught the sinit the eeeasinn and it ditntinishess eenifert and restriets f1reed en1 nth n1ev erneint if it is eixther the lease er ten tigi1t History and hingraphy There is an nld Ai n1itetniani saiytingi He whe i CilttS a sense ef the past is entnrileninetrd tn titre in the names darkness ef his nwn genetatiien Mathentaties wiitfhnut hi ii fy is rnatheniaties SilfippEd eff its greatnietss fast like the ether ar39tsampaini i7TI 39a3939t tiquotl3I lquotl i391Ii39IZ39 is nine ef the sfupireme arIs est eiviiiaatienait dveir ivens its gI i39tCiE L1I fmni the feet ef betiitig a hmfnan ereatien In an age increasingly Ei mil1 ied by mass enitnrze and hnrteaueratie itmtperstenatiit1w I take great pieas39Lir e in knewing that the 39t39it i ideas ef iTl ii 1E m iiCS 39ser39et net printed est thy a cnmputer er quot v39 tE t1i threugh 0 a CiDtt39IIt tit lIEE but instead WEI Ei eteated by the seiitiary iaher and indii 39idnal genius at a es lquotBT7lquoti3Iquot39iiEli3iE nieini The many hiegrsphieai mates in l heelt re eet my desire the enee semettthiing of the aehie39semenits and petsenai quialitiest set these astonishing human beings Metstit nf the ienger C PREFACE TC e FIRST EDmTmH nsCites are plamd in the appendic elts but each is i nk Ved dire ctly m a 5pcci v mntributinI1 IIjiEllESEd in the traxt Tihese notes ha re as thEif39 sufbjects all but a few 0f the gIquotB l39E5tK mathEmatic4ian s Hf the past Ihfi EE TE39UIiEE Fermat Negwmnj the BErnunullis E l Ef Larange Laplace Frgmrierir Gangs Ab quotPa4i5snn n Dir i1clthleVt Hamilt n Linuville Chby5he39v Harms itgpes Riemann Minknwski and P inar As T K EVli Dt wmte in annex Hf l1i5 easays S m nneT said The dead wri ters are rerntte fmm us because we kmfrw BU much Ammfe than ilthey 39 P1re cisely and tihey are thaI which we 39k39nDW ON and Vbingraphy are very uumplex and I am p aiLnfuVAly aware that scarcely anything in my mtea is actuaIll 339 quite as simple as it may appear I mmI Valsn apnk1gize far the many Eme5siv39Ely bI iEf39 a usians ta mathiemaatial ideas mars srtudernt readers haw mm Tye ncnumVtered Brut wi j the aid Of a gtmd ibl I3939 su1 icicntly inlcrcsted st7udle t rSh 39Ud Ewe abl tn unravel mast Elf ham Em them5ewe the veryquot least sum efforts may help to a 39fE rling mt the immense diversitj f classAital malh ematicse an aspmzt of Izh subject that is altmmst Vinvisibie in the a1rerage undeIgrad39uam wurric uium 1eurjg F S z rr1rne3ns39 SUGVGESTI1T7SFR THE I STRUCT The felewquotir1g djfiegmm gives the legiieal dependence ef the ejheapteers end seuggeests e vareiety ef ways this Tbeek can be ueeede depending en the purpeeies ef the eemse the emstees of the instrrueter and the beekgmunde and nveeds ef the Skt u EI139tS39i E The Halun 1I D i 39ei I39ni Eqgumium s 5 pir ab e EQ ll393i i39i i i i 1 Fmesl Elder EnILih ii I 5cendfirdnzn Lmrsazr T 39 Eqmmun5 I2 Th faleumua crl Vanatiuns I 6 I Dualhllaniw M T FltrnprM395 I11 I iLapiau39 TransIerm5 5w39nILlquottirIaI1 5 I 5 Fewer Scrm 5uluEeiun5 and Spccilall quotI I hnrnhn1 I arr E39quI iIL rrIs NJ 53mnm er Flnin nderr kEq1uauvim 5 Flu nnIirL1 In 91 Fuu f quottJ39 Series an d Drl hcrgun 1 Fun I inms M 1 FquotE1 quot f 3 li39quotlE iquot F39I iii Eq u Mimmzns 1 nd A V eeuundauyvaavuc V F rlriem5 3 Sum Special F39Llil39IE5f El l7 1v5 ml Pm391aIhemar1ec a Fhy5ic39s LL iErn5lrnct arm H Numerical U ldL39t39Er39lF55 Mgthud Theercmi M g OF UATI O p1 p1 A EPA LE p p An equatinn inv aEvin mm dependent variable and its dErivativ e5 w39ith respe1c t 0 tune or mam i ndepemlenAt vaVriaL1ble3 CTHNEEJ a d Fermt u Eq39u i 39fL MVany nf tamphe gineraJl laws of fmatjurca fin p hysics che mistjry ibi lUgy an11 astrV nnm y ndi their most namujral exprE5siDVn in the larmguagez at diJ emntiail equati rnnsx ppli39CE3Jti39ClTIS also abmmsrnd in matlm e m39aftics itselff E 539P39 3ETi M39Y in gEmemrjy and in enginEering EltC ITI miiC5 and many nthefr sgws ezunf ppEiBd sciience It is easy ta U dET5t I1 d the reasruem b el1ind this broad utility of di Erient i aI Erquamin n5 The raasder will wrevzzali that if y 3 i5 3 gfwem functi 1n than its derivat4ive a397 uquotdxV can be imerpuretc as the rate if vlmange nf 0 T P respect In 1 In any natural Aprucesa me var iahe5A inwJlve td and 1eir rates Uf change are 39Cl I 39I 39ECtEd with DITIE armther by means Eff the basin 5ciEnti E princip7iue5 t ha1I gnvxmrn thr pruwgsg when this C I139ECl i39D expirecssed in math mamical Lsw39mbn s the Eresmt is aftem a dAi ErentiaI equatiun The fo7llDwimg exampeE may illumiquotnate theae remarkza Acrgt rdling In Newmm S f d law Qf mm ti Un4 the acceflEraxli1n 11 Inf 3 had Bf ma a is Zl1 ZplTI H 1EIl tn the tum f ICE arfLing an it with Wm 13 rthe mnstant of 1 DE FFEREtfI39jAlL EIQUATIQNE p rteperttietntatity tie that H FM or me F 1 Supptose for instance mate 3 hedge et mass m tam freely under the in uenee ef gravity altenet In this case the enly free eetitng mt it is mg whetre g is the eeeelerati Un due to gravittyquot39 If y is the distenee deem tn the hedy from 5eme xEd j1eigh tt tLhen its v eity TU 2 dytfdf the rate e17 change est peeitien and its eeeelemtien e AdvMt ie the rate of ehanget ef vetleeitty 0 this tnetetiern 1 becomes art mm dfquot dz p E 3 git we alter the situattJi en by as5uming that air exerts a I ampfStiS ting feree prvperttienal to the veteettyr then the tetat fetree 3tt39i39ttg en the body is k1 yfd tt and 1 beeemes i quatieans end AIE3 are the di terentiattlt equetieens that express the eseentitel equotttribtutttes of the physiealt ptretetesstes under cetnsetitertettitetn As further 39BZi I39 PE5 ef di fetrenttitatl eqguet titens we list the teltletwtmgz e W Ami Ry 5 E E 6 t56tamp in i dz it V 1 I2 7 quot MI U U 3 dgy d 0 V t j The LilBpEndE t evatr39iebtlte in eeelt trf thesge equafitm5 is 0a and the independent tveriatblte is either er ex The letter3 k 00 ate p represent mg een be cen5idetedr ctm5tent en the surface rm E the teerth in meet apptliteat39ite ne etnd ep39ret39tittteteIy 0 feet per eeeentd per seettrtdt er Q30 eentimete Ir per E39EDvIquott pert 39E39E E Tm39d THE 05 5 ElU39AjTlDN3S E EPARABLEl lE ll tTlrDllE mnstalrtts All trdlmry di f39E Efv l Eqlltlli lt is ne lam wltitl391 tlwre is 39D39lquotquotIlr ante lnludlependeltlt V lfi bl so that all the ElEI lllquot3ILl39ll39E395 muztlrtrngg in it are mElinarjr ldsErivatllves Ejacll f lil39lESE eqtlatl ms ls rdinaryl larder 11tf a dit lerelntial IEqlll lfZl l l is the mder f the iltlgltest dEirivatltr el pl39B5E 39lf Eqttattil rts B and 6 are tst tletder erq39ulatl ns arml 39l39l39lE tll r are SE C l llCl tzalrderl Equatllonls B anti 9 are clatsslcali and are clallledt LegErtdre qtmtlan and 39EEEEl 39 EqH lEEEl rH T EBB39pEEtlllVEll Each has ea lvatsrtl liJteIa ture and aa lltisatmy I E Elf llIlg btacl l lfl1IquotldlquotEt2lSl Bf l l39S1 we sflmfll study all nf ltllmse elqllatintns tn detail latetz T p 39l li l dl ErE Ff l EqH fl is UITIE iIwntlving l39l7l39 f39E than one lindependelnt varlalbltet EU that tlm detilvaltives tll llrfl g in it are partial dl etlilv atlivtest For texamgpl if w Z flxylz39l is a f11l39lEtfltZt13971 of time and the three recttamgulllar E rDf39Elll39l tZES of a point in space than tzhet fmtltlnwlng are partial dl lEl39ElZ ltJl ll Eqlualtllmts of the 5emn dl tnrd Er alw Slew SEW 353 0 2 992 321 W E x T 321v E2w aw 25 Elw E 33 3 3 532 t31t2 These ecqlttatttll nls are lalsu cllassitcal and are called L pl39ace 3 ualtlal the htmt equlatEanl an tll was elqluati n t e5pecltiwlyl is pttrt nlulnrdllyl slgnll tclantt in tl1e At etitall plilysigcsl arld their study has stlimtulatad the dIEllsBlt pIquot1 TIEatlll titf many impurtantt lmlathlEjmaltitcal ideas lint genelral par39tial dl erenitiatl E39ll 39l l lI39llS arise in the pthtysitcsl cnntinurzms mealiaallt1 rblltems invUtlltrltmig electric elds uid dynamics 39lll lL1S lUl39l and wave l Ili39lZItl Dl39lg Their tha ry is very di erent from tl39tati Elf Dtrdtllmtry dai relntlal Eq1Ll IltZlllS andt is 1l39l39ll39lt3l l mare di T39lcttll39t alttncrst eramy tquottiSpErCl Elm 5Izrrmal time tr zz mc we shall cun lwntei our asltt ntlillnltt ezxgclluasilvaly tn ordinal y cli lertentlall equaIiun5E 2 The Elf39lgll5l I hlttltingist J E 0 HaldainE l3 9 2 jltas E gaind rematlt ahtlut the nnelt dimenltsl1nal 5tmlaLl case of the heat equatltrr1 In sclenti t th uug h t we atlnpt the si1rt39lyplaE5t rthwry will Exlp aliln all the fsacts llJm2lEl39 can id1r39atllDnl ar1a I tenable us to FlledLi7t new facts ml tilt lsame lltimtj Tlte catch in this crit39rlunt lit5 in tlt2 wUrd silmpln5 t It ls r alljr an acsthetl mtltm such as we nd lmpll lrlt ln ttt c rltici3m mil pn etnl Gr paln39llttg That lawman t nds sutch a law as 3 32w H e E M at much E55 simple than it mazes raft lwhlcih it is ll 1 ma thtmatlcal 5 talmEn39tl The 39phj saiiiEt lElt 39El39iSE393 llll l juclgmelnt and fhtt 5taleltt i1t is El t39l il391lji39 the mmquotE l39rttilfui ml tlfls 39Iw ED lazy as ptEt licti n is E nEtrn E d It la hntwzevm a 5tatEr ttvEn39l ab39l39ut s mEtl1eing very uln ltmilia r to the plain rnan num lr39 the rate at uzhanlgc at a rate trul cIhangcquot DlFF39EFaEHT39M39L Ema1fisHs The gET lE39l397 i mdiinary Ciii i393I EIlii i i equatiiiiun of the nth mrderzr W 06 Wu F61 y U or using ti39IE prime niumiiii iii fur di aLrivatiiiv39es Fmui y 39 z a i y i 0 Any adequyaite thecrretiacal dii5cL1x5sii n tnf this Exquatinn wnuld have In be ibased men a mmfiull zstuiidy f iexp liaiitl3v ElI5S Li 1Edi pmptrties waif me iumztiun d H we ver undue emphasis on the fine pCIi tiS nf theory ftmi tends to hscum what is rarai iijy gaiing n WE will iiherefnIE try in awmiid ing ivmly fli z jf abuut 5L1ch im239aitars at least for tile prresm1t it is nurmaliiyi an siimilie task in vaiif y ithat E igivieni funct39iDn y yrx is a SDi39ILI39tiiUII i ii an EqlJ39 t QV n like All quotmat is neiceissairy tn mmipute tiimi tiaEriviatii wE5 of yx and In Shaw that yix and tghese dE fiiquot39 39 Vt39iVE1E wihien sub5tiMi tEid in Iih E qL1gati Jni T d l iCE it In an irdenitiitiy im 1 in this way we 39 iE that 2 El andi is is 3 are bath si liuti mins rif39 the secnnd Urrdser equati n 39 5y fry 2 0 andia imam Mgenerially39 that y 161 353 is aisni 51 soiiiiuiitimn fm Eveiry ChU39iE IE Bf the E n ta t 1 and Simlutiiuins f diffxerieiinirial equati ns tiften arise in the farm of funminns de ned imVpliicitiyi and nme timis it di cult Ur iimjp ssihale In eIprEss the diepienidianit variiiaibiie expilii uitiy in 1e1rrns 1 the imidiepeindenit 39V fEi i J iE Fmquot iiiinstarice 2 iiugy c i4L is a is Dfl39uitiinn Qf 53 Mi dx 1 as xy am far esweiry wavn1ae mm the mnstanit 3 as we can readily verifiy P di ieriEntiiiatiing 4 and marraniging ithe rB5ultii3 Thiiese eixaimpia3 aim 3 in c aicuIusi ihic nuiI39aiim1 in ft iil wsad fur the istwizialiiud numm irrvgiari iimiha I Him 39funrti n itgr 3 In m m adv39anti3ti m3ur5 E hUwrwEr Eihi5 f39LwI1 iinn is ii i xi a wa395r di1md by the ijvrn laui iwarg J5 MATURE DE z E 9U39ATli39H EPARA E LE EUUHTJDNS 5 iiiiiJu5iti aitn the fact that a 5 i1lIii Ii of a idi ierentiiiai Equaiiiun uiswally cmtains mine at mme arbitrary l3UFiS t I itS iequai in nuim bier to the ecrrdcr of the equa uni in must BEBES prmcediuires at this kind are ieasy tin apply to a suspected sufiutim of ai Jeni idi erenitiai equatiuni p8 p139gt0bLiEm of stiariiing with La vrii quotenrenitiiai equmun and i ndiitng 3 s0 iuiIim1 l39l 11lIquot ii397 much mare di icult In dime 1nursei we shall di Eiquot F iliip 5ysitEmaitLic miethtziudiisi Em smving equatiins like and P AFQii tile iJITE5EI1ti h WE1aquotEr iwe iiimiit cwurs EIiv e5 in a iiew remairks an some of the genera asperzzis Di snlLMim1s The simplest uziaf all di iereinitiai equati ns is and we SIJIVE Ab writiimg Jrfi1 c T in s cmiei Cas e5 k ine niitei ii7i i EgTl Cli T i39i can he wuIriked mil by the meIh dis Bf 1raiicimiius In 39llquot39mer casees may be idgipcuiiimi Ur implilS5 iiilE to nd 3 fnrimuiiai far this integmi It is knuwn fur iimsiancieii that 39 i Sim J e quot1dx and P cannui be expressed in terms Va miie numb r 1if eiemiieintiairy Liurictintn5 If we recall hweciveri w I rm l is imereijw a Sjf ibr i far 3 I CtiD n any fiuncti m with derivaiiva M then we na11 almost agiways give 7 a valiid im aniiinig writxing it in the farm y Jxfr as 3 Tim crux at time fEI7IEiIl Ef is mat dE i itE integri ai Z 3 im1ctiDm lcifi 1113 uppar limit 1 the I uncie1r Ii39ic ii if gf ii Sign is my 1 dumzmy 39ie39 I i h1iB iAny l39Il39 3 N3F whiz t Kcuiriu uis iabwzjaut rtlh rEa5nm5 i if mas 5huruiLi mnsim D Maiad quot 39iZniegra39tiun Ami Mam Manuriiy mL 63 pp I152 zJ5 1 i391 Fur diiliiiri ii s i dE39tail5 ISEE 01 Iii H39ardyu The i39ntf39 gra ian inf F Ei i 5 av S7in1ga39E39 i a rix391 riquot 39CamhridgE Uniiaquot ir5it 3quot PirquotiEi Lcjmi nx 1916 Dr F Riflii fHI39EgF i39i 39H in Fi39nim TE FFF I y iuiumhiia iUnimeAr5it3r PrEiEsii New Wnrk ISMIEi 6 nEHT1AL 39EQ Ua1 IcH S which azlways exisvts when th interand is co4ntin1mu5 cwmr the Aruanxge mf i39ntegrVa1imn and that derivar i39ve is Vir5 The snwcafli ed aaepnrabie eqmtimns or eqVuatinns with zseparalnle variabl a am at the same llevel uf simplViecity as 6 Th s are diffar EntiaI eq1mtiampru1s that can be wrrittxen in the farm fix wl1ere the trigm gide is 3 prm Juct mf Atwu functiarns each of whiEh d7epEnds on only one nrf Athe variahltl5 In such 3 case w can s epa4rate the variabil ts by 39WI itii g gy and than sashei the miginal EL1I3Ii by integrating39 Jfxdx 2 Th se are sAim pIe differ nt4ia EqL1339IiampEJ H to deal with in the SE39FIE39E p the parucublam of sf1 ving thciem vzan ht rcdu Ed ta the problem Bf integratinn Even Eh llgh the iAn di1 catctl itntegratimms can be di icmt DI impcassijbley to mm39 om exAplVicitlly The genairal rst 0rder equati n is tr HE 5 pEti i case of A1 Vwhmh cor rEspUn d5 to taking m 1 dr Fm 3 8N 9 We n0rn1naliy expect 11 hat am aquaIim1 like this will have a srnlutirL and that mire sDluti rnelike T and E wiM crmtain sane aVrb iILrarr E2DIISIHHL 39H WBquotvquotEi39 zI R 2 U 1Lr has no real va14ued 5ulutmnE at 31 an 7 ail 2 G U tfr quotF has 39211 y the Si g E s0luIiaan y U which unntai n5V nu arhitr39ary mn gE iI1tJ i S iquottuatirJns f Vthis kind raise di f cu1t tha0re21Lial c39guestions i ibI i3 L1I E This s LatEme E ps nine Bfmm turf the fundamnmI mt r39em nrf 39CEIiCU U39iA THE NATURE EJF DIFFERENTAL ECJU 1TE39EJH5e S EPAR LELE EDUAT I NS the EH13tEe EEs and nature ef ee llutien n5 ef di39ff39E Ll39BIti1l equeatieeme We eannete enter here into 3 mm diseussien ef tehese questien5e bum M may eflariefy mett ere if we rgiwe an in39muietie39ve eleseeriptien f a few ef the basic feetee Fer the sake emf simplieite3r let us aeesuemeee 1hat 9 eem be eSUVEd fer if I dx We also aseueme that fxyr ie a eeeetieneueeue funeetien threugheuet same ereeetraengle R in the ey plane The gvee meeetri e meaneing eff a S lLl39lEi enf 1 eatn beert be urmesreteed es ferllewe u 1 If P e x JJ39 is e peint in R then the muembeer Q x T ffEI1wJ tI Pn de1zermin ee Nae dieeetien at Pu New let P p xhym be e jperint neaxr H in this dieeetien and use 8 Z 8 dx Ate determine a new direetjen at F Next lei P1 e 3iy2 be ajgp739DiI1 t near P 31 IF 1 reey rm 3 DIFFEHEHTIAfL EeUeATIeeI395 in this new direeeteiean emi use the number 3 39te deeermine yet enwther dixeetien at P1 If we eentinue preeeee we F ebtein e breken line with peime eeattered aleng it like beads and if we new imagine that these seuuees5ive p inte mcwe eleeer Ice eme another and beeerme mere mumeereus then breken line eppreaeeheee e emeeth euerve thre11gh the initial prnin t Pm Tliiee emvee is a seelmieen 2 Vy x ef equetien ul D fer at each jpeint xy n it A slepe is given by fjryIEemrd quotthis is prireeeieeley the eeendieteien rvequirede by the di erieentiel equatiene If we etert weith a di ereeent eienietieall pemt then in general we ebteien e 1l iEfeerent euwe er SDlU 39inj Thee ee 1l1 tiene ef 1031 form a femi egye ef curves eelled rueegerei eumeesf Ferthermeree appears 1311 be a reeeeneielfe geese l rh lt threugh eeeh pevim in R there peeeee jest ene imegzml eurve ef 39die us5inI1 is int e11ded only tee Iend iplausibileitj39 tee p fellewing precise eteteemem p Theerem 0 Piear s tTheer emjl If fexe39 end fay ere eeeequot39m eu fueee39eenee 1 e e39 e5ed ereekexengfe Ff thee Ihreegh eeehr pmeiet xueyn in the eiHfEFf F r3f R mere eee5 e ei meiqee feI39egrm39 eeuree ef the egtqueIiee dy f E If we eensideer a xed value eff 13 in tfhiee th eeremi then the inteeggrael curve the passes threeeuegh fxmyuj is fully de39t ermined by the eheiee ef J e In this way we see thet the integrel euerves ef gtl ceneteiM1te what is called 3 oneeeremeeer family ef eurues The eguatieene ef this familjy we be wermee in the term 4 yireE 11 where di ererent efjheeieee ef the pearemeeter e yieele1 iefferent eun39eee in the efemilye The inte1quotel curve that passes threugh xmy i eeerreepende ete the veleue ef e fer which F xe3e If we deeeetee this nuembere by em then 11 ealleede the geeneref se erEeen ef elk ere y yixaecol is eeellled the perrieuier mfuimn that sateise eee the im Iief eemafiere er1 O 2 y when x E Se i1L1tisene eE an c ffereneiieeam sElLil3Ii 39L39I are eemeitimee ee led r39jnegruxi 3 ef ithe eque tien beeeuee the pereeebeeem eF ndirng them mere er ees en eeIee5iue rn mI739 the ersd1iner3 preblern er in tegreI39i ee NATURE BF p EQUATIONS VSEPFLRAIELE EDMATIUNAS The essential festtire Df the general seit1tien 11 is that the eenstsnt e in it sen be ehesen se that an integral eurve passes tthreitugh any given paint the rectangle under ieensiderstien Piearid s theorem is prenred in Chapter K This preef is quite eremtplieetted and is pmbshiy best Apiest gpnned until the reader has heed eewnsitiersbtie experience with the nmte straightforward parts of the suhjeett The theerern itself can be strengthetnted in trerieus direetiens by weeltening its hyptheses it can also he getntetsltizieidt te refer the nth er de139 eqnetiens seihrsih iie fer the nth etdeir aierivettitve Detailed ideseriptiens if these results wetsuit be eaut eff piece in the present eenteztt and we eentent enrseltres fer the tinte hierigng with this inferntel ditseuss39i0n ef the meitn idessi In the quotrest ef this ehepter wet eitplexre some ef the sways in whieh di erential equetiens arise in seienti39 e epgpiieetti entst PR B7LlEiMS PI 1 that tthe fezllewing metiens gejtpiietitt er i39 39lpiiEi39U ere se1ntiens ef the eertetspending dti erentiai equetiens e e2 It y git E I By p 3 e1quotquoti e yyquot e39 di P J kyiz e1 7 sin 2 egeersfzx y 4y U f eteE e2te Equotquot y 393939 4 0 g s cm Ei hlt E3 39D5h P 39 4y U M V p s it1rquotquotI3 icy y yquot Z xiyzg ii 1 t enx xy y 753 ya it W lees W y T V 162 es 262 ya 3 M 3 teiquot vr39 amp 39 ut39e er MI Iquot 41121 yi 2 ft efx y 19 xyi tiy f 111 y X si 9 Wtxiyi xi i39 siny st eesy39 sinty xvy pU e 1 tenquoty 1 y 2 i A 0 2 Find the general SDi1lItiiElI1 ef each ef the fellewing dfi ieIen tiei eqnettiensz H J 0 Ar Iii xisi yi 039 bi xw 1 it ty3939 2 1 zsman y E 3quot Isa U J A y sinquot1 p2 y shiny xi 1 txky39 2 X quot rt h sine 39 1 f 1 x1y r er 3 tense Ii is I W F E 1 Eh 1 tEyquot ten 1x q 1 e dy is hr 0 ti sissy y I F s i isydx me at Fer esmh ef the feliiewini diffEI39Efit iEi equetienst nd the pertieuiert setn tie nt H tl 10 ErIFEREH nAL J239UU JI39lCINS that E jtizs th gimczn initial candi tic1n re y 2 when I ll 39 3925ainIc 1151quot y p 1 Wham 1 3 O 9 39quot T 103 y IE1 when 1 W E quotdim x 1quot A 0 y z B when 2 m E 39r E 41939 9 y J wh m 0 E 1 P0 1LIrT3 1 12 I when x U P Fm each of the fall0wVinAg d ierential Equa39tinn5i nd the intcgral cum that passes thmugh the given pmin t H J 83kgquot Dr W Eh xi 1 L U 0V edx 1 girl dy PI 1 p c39cws 3x39 105 2 PM E sin 3 sin P E i39p 239r 3 2 0 E3 was 9 quotElU 0 IE 1 my L 1 aim Slmw tElml V E 2e dr is a 5fmtiun 21f yd 1 Fm the d9iearEnIia1 equaIi10n 2 namelyquot Pquot 53939 6 A carry nut th detailed Cai ulaliDRE n eeded in verify this as5e1rtrinuI1 in the ttv39I1V t hat aft 3 E1 and y E are bath 5mnlVu39tiun5 and b y 2 Ejy h EEIESJ 3 smutim fur awry th irt of the constants cl anrd C1 REma rk In Studring a 393nnlI like rhi a 5m dent 5hQ ud E quotUE39f silide past a5sertiuns mt this kind inv0lvin g smzgh phrases as we 5amp6 GT as we can readily 39mrif39 wit h u t pcr5 na Ilr IhEzIking 39I Bi1quot vallidit r39 quotFhamp mem f t thaIE s wmething is 7611 print dne5 nut n1ean it is necessagrily true Cultivate EkEfPITiEi5 1 as a hea1 thy smte mf mind as rnu wnuld phjmical tnes5 accept rmthing on the auIhmquotity39 mi this wtriter or any urthcr nutmil yarn Imusre und rsta d it fvully mt ynurzglf P In the spirit of Pmbl em 5 Verify that 4 a 5 l11ti n iif the iEerEntiaJ aquarium 55 for every valmme uf Elm cm15tgtant E 9 Fm wlhat values of the cnnstant will y 3quot be 3 5ulutimn f Elm dii eIcntiaIl eq uatiDn 4115 the ideas in PmbIem 6 tn find 3 5n1 1u Im n unntain img three arbitrary L We have SEETI that the gAner al 5nlutinn Elf 3 first rdeI di FfE I intia equatrmn 1rmrmalljf rmtains one arbitranr E 39nJEt3nt called 31 parameter when this parameter is assignBd vay1rim15 y HBE we ntain a armc F 1EIElquot family Df ciunres Each of tlms3 rurves is a par39ticular SDlL1tiD r integra4l ruWe at the giv En rliffsmntial equatim1 and EH Hf them tcu ge39ther mmstitut p genreml 5nImi0n THE NATURE or DIFFERENTIAL EQUATIONS SEPARABLE EQUATIONS 11 Conversely as we might expect the curves of any one parameter family are integral curves of some rst order differential equation If the family is fxyrC 39 0 1 then its differential equation can be found by the following steps First differentiate 1 implicitly with respect to x to get a relation of the form 0 8x dx 3 C quotquot Next eliminate the parameter c from 1 and 2 to obtain Fxy a 0 3 as the desired differential equation For example i x2 y2 C2 4 is the equation of the family of all circles with centers at the origin Fig 2 On differentiation with respect to 1 this becomes d my m FIGURE 2 12 DIFFERE39MTIJ 1 Ff3vUATEiHS and since c is a41read1y absent there iamp zrm neezd t eliminate it and xy 0 mm is the di erenlttial eqJuatin f the given famxily mf Ei139ElBS S i milarl3i xi y 263 2 is the eVq uati1m eff the family Ellrf all ciml s tar1geVnt ta the yaxi5 at the m39igin Pq p when we d i7 7eruEntEiate tl 1is with rB39ipl c39t tn 1 we Qb1ain dy 3quot or I 0 C F Th param ter Still pjf l nl SD it is n em55ary TED amplimAinatE P by curnbini11 6 and 7 This yields 2 T 3 ziij Pp as H16 di e1EnIiaE equati m of thv 39faII1iIl339 6 in 3 THE NATURE or DIFFERENTIAL EQUATIONS SEPARABLE EQUATIONS 13 As an interesting application of these procedures we consider the problem of nding orthogonal trajectories To explain what this problem is we observe that the family of circles represented by 4 and the family y mx of straight lines through the origin the dotted lines in Fig 2 have the following property each curve in either family is orthogonal ie perpendicular to every curve in the other family Whenever two families of curves are related in this way each is said to be a family of orthogonal trajectories of the other Orthogonal trajectories are of interest in the geometry of plane curves and also in certain parts of applied mathematics For instance if an electric current is owing in a plane sheet of conducting material then the lines of equal potential are the orthogonal trajectories of the lines of current ow In the example of the circles centered on the origin it is geometrically obvious that the orthogonal trajectories are the straight lines through the origin and conversely In order to cope with more complicated situations however we need an analytic method for nding orthogonal trajectories Suppose that if fxy 9 is the differential equation of the family of solid curves in Fig 4 These curves are characterized by the fact that at any point x y on any one of them the slope is given by fxy The dotted orthogonal trajectory through the same point being orthogonal to the first curve has as its slope the negative reciprocal of the rst slope Thus along any FIGURE 4 14 emFEesrrmL seusrrtesss ertltegteneil treji eretery we have dyf s1fftxy er f miyit mu y Our mettihed ertt Ending the ertitegnat trajeeteriesi ef at given family est eurvzes is 39U39IETI fm39 E as feiiews irsit ned the di ferential equetien ef the fsimily nest retpillsee dyfydx by dx e39y t obtain the di erenitisil equstien of the lquotti39lUg Hl trsjaeeteries and i nstlly solve this new di fereimtisl eqsuattien If we sptys this method ten the fsmi1ty ef eireies 4 with di etetttial eq39IuaftiUt1 5 we get G1 e Jill as the diEe reantisi equstieln elf the I3tth gexnal trsjieeteries We ean new s6pBt39 t1E39 the Variahies in 11 te ebtain dye 139 J Whiehi ens direct in39tegt39stitm yiieids logy leg t 10 ct err is ex as the equeti tt ef the erthegenal 39trsjeeteriesi It is etten eensvenient the express the gixvein fsmtily of euwes in terms if peels DtJI iiITlaEBS In this ease we use the feet that it it is the snglie fIquotD II t the peter radius ti m tangent then tan 1 r d fdr By the sleeves diseussien we re iseei this EtpI39E5 i l39l in the eIifEet39entiisil equstien of the given family by its negative reieipmesl drtquotir d to Ubtssin the di erentisl equstien ef the eitrtsthegeanisl trajeietieriesi As an ilhtsttrstiien ef the value ef this teettnique we nd the etthgensi trssieteterties of the iiamisihr f eireles 7k If wet use reetsttguler enervdins tes it fe erwtst frenm f 39 39JIa t the di eret1tisi Bql I IitZt ef the erthegetnaii tmjeeteritesi is 13 is 33 yr 12 U fUftM39 tvEi 3 the vsristzrles in 12 C I tt39ID E he seipstetiedi se witheut scIdi39tiena1 teehniques for selviitlg diifferentisI equatiems we seen go no THE NATURE OF DIFFERENTIAL EQUATIONS SEPARABLE EQUATIONS O yl HV FIGURE 5 further in this direction However if we use polar coordinates the equation of the family 6 can be written as r 2c cos 9 13 From this we nd that dr d6 2c sin 6 14 and after eliminating c from 13 and 14 we arrive at rd6l cos6 717 sin6 as the differential equation of the given family Accordingly r16 sin 6 dr 5quot cos 6 is the differential equation of the orthogonal trajectories In this case the variables can be separated yielding c1rcos8d6 r sin9 E U39eT1 fH5 and sffter inteIatin this eeemes leg rs legsin P lLeg2es sue that r 5 2e sin 115 P the Eq i1ati 39 ef the etrthegens s tIan jE39C39t ri39EISis It will be nete d that 15 is the equstien ef the famili3r et eIl eireles tangent tn the retxEs est the snsisgins see the detted eerves in Fig In Chapter Pq we develop a rm39mhbtet ef more etlekhotete preeettuteres fer S vDhquotii g L rst erder equetiens Sihee em present ettenttietn is tiireet e d mere at spp tisesti eents than fterrnet VEBTCht Kltt39ES all the pt blems esimen this eheptere here seilevefhle p the TTlBIhJd ef seps139etisun ef variables ilhnsetreted shove 1 Sktexteh eeeh ef the E eHee wing femiliest emf teuwesi nd the e rthegeeel 39trsjeetet39iets and eelijd them tee the sketeh s xy 2 e fejn r erjl t ees C ttt p y ccquot P What are the eI theget1e 39Irejeeteries emf the family ef eruWes st 3 eJiquot Lh I y er wheres r is any gpesitive imeeger t In each eases SKEIEJII hem families set curves I39e57t st is te emst en the ertheegenel 39trejee39ttTmies ef increasing the Ex P 1n t H n Sheer tether the method fer t ndtng erth egenal trejeeteries in peter eeerdinetest een he expressed as felhjwst If e r39d Fr is the citfetquotet1tieJ eqIL1etquotirrI1I ef the givens fsmisly ef euwess thein drld t rEfFhr 5 is the di erentie equettien of the ilftth g l ettejesetteri est tIJfpl4quot this msethee te thee tf7em39ils ef eireles r 2 sin 9 2 Uste peter eeetisnetes te nd the er39thegentst1 trssj eetet39ies of the femsily set perebelias r effl test 9 esquot 2 Sleeteh berths f fmml of eursres pg Sleeteh the family 0 y s 47es e ef elil lpetrsbeigisetsx with exits the 1 iiti ends fereus at the e rigin t39It2l nds the differential euettisen ef the fesmilys o that this di 7esentitsI eqtaIeti eh is una39tt eired when tdyset is repleeed It 2 It eet1elustien cent be drawn fireasm this feet 5 Find the eerves that satisfy eaeh er the fe ewimg geemetrh eendiittiehs e The part ef the t3Iquotlg39BI 1t eut eff hy the ashes is bisected hy the peirtt est ttengeney Eh The pmjeetien cm the k ef the part ef the netmel between y am the x esis hes stsertgth L e The ptejeetiert en the retie ef the part ef the te get391t 39betw een h and the Jsaastis has Iength L i The part ef the te11ent 39hetfwe39en 39y senses the I i g i is hiseeted by the ysx1ts THE NATURE OF DIFFERENTIAL EQUATIONS SEPARABLE EQUATIONS S e The part of the normal between x y and the yaxis is bisected by the xaxis f x y is equidistant from the origin and the point of intersection of the normal with the xaxis g The polar angle 6 equals the angle 1 from the polar radius to the tangent h The angle ip from the polar radius to the tangent is constant 7 A curve rises from the origin in the xy plane into the rst quadrant The area under the curve from 00 to x y is onethird the area of the rectangle with these points as opposite vertices Find the equation of the curve 8 Three vertices of a rectangle of area A lie on the xaxis at the origin and on the y axis If the fourth vertex moves along a curve y yx in the first quadrant in such a way that the rate of change of A with respect to x is proportional to A nd the equation of the curve 9 A saddle without a saddlehorn pommel has the shape of the surface 2 y2 x2 It is lying outdoors in a rainstorm Find the paths along which raindrops will run down the saddle Draw a sketch and use it to convince yourself that your answer is reasonable 10 Find the differential equation of each of the following oneparameter families of curves a y xsin x c b all circles through 10 and 10 c all circles with centers on the line y x and tangent to both axes d all lines tangent to the parabola x2 4y hint the slope of the tangent line at 2aa2 is a e all lines tangent to the unit circle x2 y2 1 11 In part 01 of Problem 10 show that the parabola itself is an integral Curve of the differential equation of the family of all its tangent lines and that therefore through each point of this parabola there pass two integral curves of this differential equation Do the same for the unit circle in part e of Problem 10 A 4 GROWTH DECAY CHEMICAL REACTIONS AND MDKING We remind the student that the number e is often de ned by the limit 1quot el1m1 rz gt00 71 or slightly more generally put h 1n by the limit e lim 1 h quot 1 hgt0 In words this says that e is the limit of 1 plus a small number raised to the power of the reciprocal of the small number as that small number approaches 0 3 resell fmrn eeleulus that Itlhes imspertemee ef the number e ilies mainly in the feet that the e39spenenstiel fune en e is unsehengesd by dis esremtissiEesnss if e gtTquot39V gg An sequivelent statement is that y e equetimquotu I 1 is e sEu1ien ef the sdi ereenstieesl e ex t More gemereM3rssL if k is ens given sneeeesre eensstsm then all f funetimaas P8 3 see are selustiens ef the Ji eremiel equetiem g This is easy Le verify by dki ezressntistsiesni end seen else be diseevs ered by separating the quotvariables and isntegreting p 1531quotIter r Jet F nL 39 may p1 Em sgv er 2 as Further it is net diq ieuslt in sheer that hesse efunetiens are these esniy slmiens eaf equsetie ns 2 see Preeblem g In this sxeetitm we discuss a ssurpsrisinegl3r wide versizetyr of epplieat39iscmse eff these ifeets tie ee numfber ef di ereeent seie11ees o o H 2 Esamiple L Celslinsueusiliy inters39l If P d liers is deepeesited in a beam that pejFs an int eltrest rele eff 6 percent per jeer eempeundeed seemisnnL1elljrs then after is years the eeeunf1u1st ed emseum is A P1 003quots Mere ge esa1Iy if the il Ji EIB5t lquot tE is l k percent k L06 for s psereem and if this ietereset is eempeun ed n times 3 year 39EIhiEI39ll efteer I years the eeeun1uieted enleum is r pvD pvD er A as P1 J H If n anew iinerees ed inde n itesIy se the the interest e Dmpeu nd ed meme and metres frewent yr then we eppreeeh the iimitieg ease ef eeszstisnuesusly eemspesunded interestf T End the f39er39muie fer under these Bil VT Many banks peg interest daily which ee39I rsespe39Imjs tee H 355 This eesm1wer is ierge enm1gh ice melee eenteinsueesljy eempesumiled quotinteressl e very eeeuaa39Ie medel fer when eetuslly happens THE NATURE OF DIFFERENTIAL EQUATIONS SEPARABLE EQUATIONS I cumstances we observe that 1 yields nt nk kr 32 wgt l 0 II A Pek 3 S0 We describe this situation by saying that the amount A grows exponentially or provides an example of exponential growth To understand the meaning of the constant k from a di erent point of view we differentiate 3 to obtain 11 2 Pkequot39 A If we write this differential equation for A in the form dAA dt k then we see that k can be thought of as the fractional change in A per unit time and 100k is the percentage change in A per unit time Example 2 Population growth Suppose that xo bacteria are placed in a nutrient solution at time t O and that x xt is the population of the colony at a later time t If food and living space are unlimited and if as a consequence the population at any moment is increasing at a rate proportional to the population at that moment nd x as a function of t Since the rate of increase of x is proportional to x itself we can write down the differential equation dx kx dt By separating the variables and integrating we get dx kdt logxktc x Since x x0 when t 0 we have c log x0 so logx kt log x0 and x xoek 4 We therefore have another example of exponential growth To make these ideas more concrete let us assume for the sake of discussion that the total human population of the earth grows in this way According to the United Nations demographic experts this population is increasing at an overall rate of approximately 2 percent per year so 8Brie y this assumption about the rate means that we expect twice as many births in a given short interval of time when twice as many bacteria are present THE NATURE OF DIFFERENTIAL EQUATIONS SEPARABLE EQUATIONS In x xlt V FIGURE 6 This function is therefore the solution of the differential equation 6 that satis es the initial condition 7 Its graph is given in Fig 6 The positive constant k is called the rate constant for its value is clearly a measure of the rate at which the reaction proceeds As we know from Example 1 k can be thought of as the fractional loss of x per unit time Very few rst order chemical reactions are known and by far the most important of these is radioactive decay It is convenient to express the rate of decay of a radioactive element in terms of its half life which is the time required for a given quantity of the element to diminish by a factor of onehalf If we replace x by xo2 in formula 8 then we get the equation 2 xOe kT for the halflife T so kT log 2 If either k or T is known from observation or experiment this equation enables us to nd the other The situation discussed here is an example of exponential decay This phrase refers only to the form of the function 8 andthe manner in which the quantity x diminishes and not necessarily to the idea that something or other is disintegrating Example 4 Mixing A tank contains 50 gallons of brine in which 75 pounds of salt are dissolved Beginning at time t 0 brine containing 3 pounds of salt per gallon ows in at the rate of 2 gallons per minute and the mixture which is kept uniform by stirring ows out at the same rate When will there be 125 pounds of dissolved salt in the tank How much dissolved salt is in the tank after a long time If x xt is the number of pounds of dissolved salt in the tank at time t 2 0 then the concentration at that time is x50 pounds per gallon The rate of change of x is dx rate at which salt enters tank rate at which salt leaves tank dt THE NATURE or DIFFERENTIAL EQUATIONS SEPARABLE EQUATIONS 23 13 billion years and yet another preferred for dating the oldest rocks is based on the decay of rubidium into strontium with a halflife of 50 billion years These studies are complex and susceptible to errors of many kinds but they can often be checked against one another and arequot capable of yielding reliable dates for many events in geological history linked to the formation of igneous rocks Rocks tens of millions of years old are quite young ages ranging into hundreds of millions of years are common and the oldest rocks yet discovered are upwards of 3 billion years old This of course is a lower limit for the age of the earth s crust and so for the age of the earth itself Other investigations using various types of astronomical data age determinations for minerals in meteor ites and so on have suggested a probable age for the earth of about 45 billion years The radioactive elements mentioned above decay so slowly that the methods of age determination based on them are not suitable quotfor dating events that took place relatively recently This gap was lled by Willard Libby39s discovery in the late 1940s of radiocarbon a radioactive isotope of carbon with a halflife of about 5600 years By 1950 Libby and his associates had developed the technique of radiocarbon dating which added a second hand to the slowmoving geological clocks described above and made it possible to date events in the later stages of the Ice Age and some of the movements and activities of prehistoric man The contributions of this technique to late Pleistocene geology and archaeol ogy have been spectacular In brief outline the facts and principles involved are these Radiocarbon is produced in the upper atmosphere by the action of cosmic ray neutrons on nitrogen This radiocarbon is oxidized to carbon dioxide which in turn is mixed by the winds with the nonradioactive carbon dioxide already present Since radiocarbon is constantly being formed and constantly decomposing back into nitrogen its proportion to ordinary carbon in the atmosphere has long since reached an equilibrium state All airbreathing plants incorporate this proportion of radiocarbon into their tissues as do the animals that eat these plants This proportion remains constant as long as a plant or animal lives but when it dies it ceases to absorb new radiocarbon while the supply it has at the time of death continues tliesiiiiisteady process of decay Thus if a piece of old wood has half theradioactivity of a living tree it lived about 5600 years ago and if it has only a fourth this radioactivity it lived about 11200 years ago This principle provides a method for dating any ancient object of organic 10 For a full discussion of these matters as well as many other methods and results of the science of geochronology see F E Zeuner Dating the Past 4th ed Methuen London 1958 P eIieFese2srIsLreeesT1eHs erigih fer instenee weed ehe1quoteeei vegeta hie heer esh skih heme hes hem The reiiefhiittity ef the imethte21 P been veri ed1 ihyt eppirying it he the hesrtweed eff giant seqtIteia trees whe s e grewth irings tquoteemrd K In years ef life and te rfernituire fmcm Eggrptien temhs whese serge is else imewn i1fZi pEFttlEIttlf There are tieehtihtieel diTe uities but the method is mew felt the he eafpehlet ef reetsteinehie eeeureey es leegj as the eeriei s niif time ixweittvied ire nets tee greet up the eheut yteerst Redieesrhen datin hes heen eppiited Ate tiheusends ef samples end iebeirette139ies fer esrryihtg en werici rnhmber in the d7eeehs AI139tti1I39Ilt the mere interesting age estimetes are these linen wrappihges frem the Deed See serelis ef the Beer2 ef Iseieht rEE 39iE jquot feund in e esfse in Palestine and theught ten he tiest er seeend CE 1 EUIi etti 1391 yE EIrS rehsreteei from the Leseeuex heave in seutihezm Fareneei site ef the remarkable p1r ethiisterie peintitntgs 39i551t1i 9 years ehereeel fresh the prelhisterie m t1iF Elli et Sitenethenge in seuthtern England p 5275 yteerts 1311HIUDEKI from 1 three hmned at the times ef the vleetniet expiesient fermed Cretezr Lake in Oregon y i 3veers 3V I1 Squotit39E5 ef E39iE ii men itthmugheu39t the W7ersterh Hetmisphettret hteee heeh tefteti by using ipir t ES ef ehereeei ber sshcials fragmehts ef hemed hiscm heme and the like The sesuitts seggestt that htuimen beings id net erriv e in the New World ienttii EbGut the perited ef the last Ice Ages rteeugehiy 25000 years age when the ievei estquot the waiter in the eeesns was ssehsitentieiily lewer 39thten it new is en they EDUM hashes wailte aeress the Bertitnge Straits ftrem Siheris te Aieske Lk If p rn a given ne1were eeretent shew that the funetiehs y ee are the enly S iit ti tZtIil5 ef the diEfet ehtjie1 equetii nt dy tdx F Him Asstlme that ffs is e seietien ef this eqL1et ie39h end shew that F39xfe i is e veienetsnt E Supp ese that P deilers is trtepesitecll in is heeiV that pays tinttes e st at an Eli 39l39nU Elf l Of 3 pereetht eempeunded eeintinuetJteTIjan 3 Find the time T reAq39uired fer p iIw esttm ent Ate diveruhie in e39e1tIse as are t unetient ef the interest rate r h Find the sinteresiquot rate that must be ehtsihed if the Kimestm ent is ten ieuhie Q seiute in 10 yeeersm H Lihhjr39 hthe the i191 I Prize fer ehtemistfry39 es 3 eentsetqueI1c39e ef the seer deseirihed eheve His ewn seeeunt ei the 39reet hed witth its pittfelle i iIl1td eemetusi eins een he f39eundJ in his heek H fii e fb rtt Bering 2d eti iJniversity ef Chieeg1 Press 1955 3 10 25 THE NATURE OF DIFFERENTIAL EQUATIONS SEPARABLE EQUATIONS A bright young executive with foresight but no initial capital makes constant investments of D dollars per year at an annual interest rate of 100k percent Assume that the investments are made continuously and that interest is compounded continuously a Find the accumulated amount A at any time t b If the interest rate is 6 percent what must D be if 1 million dollars is to be available for retirement 40 years later If the bright young executive is bright enough to nd a safe investment opportunity paying 10 percent what must D be to achieve the same result of 1 million dollars 40 years later It is worth noticing that if this amount of money is simply squirreled away without interest each year for 40 years the grand total will be less than 80000 A newly retired person invests total life savings of P dollars at an interest rate of 100k percent per year compounded continuously Withdrawals for living expenses are made continuously at a rate of W dollars per year a Find the accumulated amount A at any time t b Find the withdrawal rate W0 at which A will remain constant c If W is greater than the value We found in part b then A will decrease and ultimately disappear How long will this take d Find the time in part c if the interest rate is 5 percent and W 2W0 A certain stock market tycoon has a fortune that increases at a rate proportional to the square of its size at any time If he had 10 million dollars a year ago and has 20 million dollars today how wealthy will he be in 6 months In a year i A bacterial culture of population x is known to have a growth rate proportional to x itself Between 6 PM and 7 PM the population triples At what time will the population become 100 times what it was at 6 PM C The population of a certain mining town is known to increase at a rate proportional to itself After 2 years the population doubled and after 1 more year the population was 10000 What was the original population It is estimated by experts on agriculture that onethird of an acre of land is needed to provide food for one person on a continuing basis It is also estimated that there are 10 billion acres of arable land on earth and that therefore a maximum population of 30 billion people can be sustained if no other sources of food are known The total world population at the beginning of 1970 was 36 billion Assuming that the population continues to increase at the rate of 2 percent per year when will the earth be full What will be the population in the year 2000 A mold grows at a rate proportional to the amount present At the beginning the amount was 2 grams In 2 days the amount has increased to 3 grams a If x xt is the amount of the mold at time t show that x 232 b Find the amount at the end of 10 days In Example 2 assume that living space for the colony of bacteria is limited and food is supplied at a constant rate so that competition for food and space acts in such a way that ultimately the population will stabilize at a constant level x x can be thought of as the largest population sustainable by this environment Assume further that under these conditions the 1339s 16 17 IJIJFFERENTLFLL E39UUATlH5 pIZIp39ittIr it39El l39I greets let is tats pt 39petttieneIl te the ptetlriset ef I and the di ieirEn1t e el J x end n I as e feeeitilele ef t Sltetzeh the graphs est this iiutttetisent Whert is the pepeletiee 39i n er39essing tttest t epidtly NILlE1E t 39 ssien fpteuees rleettetts in en atemie pile at e tste ptepertietesl tn the IJmlbvEE ef neuttens pttesent et en m i li It en neutrens are present initielly set he and eg ItE1i llfFt S ere present st tirttest try ertdl I3 shew Ih t ml quotll r M q v If hsli lei s given quantity eil rsdium deeempeses in quottweets whet peteentage ef the erigittsl smettnt quotwill he left at the End net P yteerst At the end ef years if the ihelielifet ef e redieel etisel lseis39tene e is 20 days hDW39 leng will it tekte ter 99 pereertt ef the lstehstenee te dE EtEP d it held tZiI39f wheat teeming witht glressheppesrs is dusted with en ittseetieitie heating a lltill lrete ref 201 per Ill per heerl WThe t pezteetttsge ef the grsssheppetts are still eliate tl hteutr lstet 7 UreniemAZ33 deeeyris at e rete preper39rttitenel te the emeunt presents it sit and x2 greets see pt139 EEEeI lft st tiniest ti stud t1 shew that the halflite is has leg 2 0 Suepese that taste ehtemzieel 5tllZ139Sl tIEE5v in setltttiee reset tetgethelr tit f ernt e eemp1et1ndl if the rreettien eeeuris by misses ef the eetlllisien see insteteetien ef the meletreles tn the substances then we expect the rste ef fI393Ji I t te39l391i tquot J ef the eGa1rtt p t1 til he be prepetitienell the the IimbE1 ef eell39isiens per unit tires which in turn is jeintiy pIetpeirttienell ten the emeusnts net the suhstet1eest thst are enttettsferrnedt A eltemieel if ti that preeeetIs in this lmetrmett spelled ti Eetje tid etrde r39 rersttiee and this leer elf resetiert is eitert referred he as tlte lee ef eetierL Citstidter e ee tjl et deLt teraetien in which 1 gtettts lei the eempeltncl eeItteit1 es grams ef the rs t stubstenee enti be greets ei the seeetnti whtette e is b tt T If thezte see HA grerns est the iirst S bsit t pl ESEH I tittitielhn em l b gittettfts ei the s eeertd1 ettd P it B when at ll tttl I es e isstnetiten oi the time t 2 Many chemicals diisselrwe in water at as rete whith is jtIiniljquotprUp IliDi1iEtl tn the eme unt uttdisse1s39edi etttl he the tili erertee bete39eten the uzenteenttetien ef e 5 IlltI39ElliEd selLttien entzl the ETD39ll39IZquotiEI7lLiquotil il39fJ1I39i estquot the eetusl seilL1tiee F01 1 chemical elf this Isixrnd pissed in let tee eentsitting i gellenst ef wsttlett nd the emt tt1t1t c ttittdissellsretl at time I if it IQ wheels is H etftd I e when t t rm anti f rg a the B1i I1 DilI39lJl disselved in the tenlt when the selutiett is 5E3939Ll l39 t39ECiJ titetdet ts tthe ere espeveiel5l339 intetested is atst enl seeend erdet ehemieel resetietts eri l ne e ztnueh titers teteiletl Eiit5t2UEEiCI39 ii Lines Pettirittggt ptreltebljr the grestes1 ehemist ei the twentieth EE LlLttf39 P his heeit General Citeetistry Ed etiw T H Freeteett eetl T Sen Fteneisee ltili t See pertitttler39ljr the ehepter V Este ef iEltemieel HE El2it1 E witzitzhj is Chapter iii in the Ed ed39it39ierL 18 21 22 27 THE NATURE OF DIFFERENTIAL EQUATIONS SEPARABLE EQUATIONS Suppose that a given population can be divided into two groups those who have a certain infectious disease and those who do not have it but can catch it by having Contact with an infected person If x and y are the proportions of infected and uninfected people then x y 1 Assume that 1 the disease spreads by the contacts just mentioned between sick people and well people 2 that the rate of spread dxdt is proportional to the number of such contacts and 3 that the two groups mingle freely with each other so that the number of contacts is jointly proportional to x and y If x x0 when t 0 nd x as a function of t sketch the graph and use this function to show that ultimately the disease will spread through the entire population A tank contains 100 gallons of brine in which 40 pounds of salt are dissolved It is desired to reduce the concentrationof salt to 01 pounds per gallon by pouring in pure water at the rate of 5 gallons per minute and allowing the mixture which is kept uniform by stirring to ow out at the same rate How long will this take 2 An aquarium contains 10 gallons of polluted water A lter is attached to this aquarium which drains off the polluted water at the rate of 5 gallons per hour and replaces it at the same rate by pure water How long does it take to reduce the pollution to half its initial level A party is being held in a room that contains 1800 cubic feet of air which is originally free of carbon monoxide Beginning at time t 0 several people start smoking cigarettes Smoke containing 6 percent carbon monoxide is introduced into the room at the rate of 015 cubic feetmin and the wellcirculated mixture leaves at the same rate through a small open window Extended exposure to a carbon monoxide concentration as low as 000018 can be dangerous When should a prudent person leave this party According to Lambert s law of absorption the percentage of incident light absorbed by a thin layer of translucent material is proportional to the thickness of the layer If sunlight falling vertically on ocean water is reduced to onehalf its initial intensity at a depth of 10 feet at what depth is it reduced to onesixteenth its initial intensity Solve this problem by merely thinking about it and also by setting up and solving a suitable differential equa on If sunlight falling vertically on lake water is reduced to three fths its initial intensity 10 at a depth of 15 feet nd its intensity at depths of 30 feet and 60 feet Find the intensity at a depth of 50 feet Consider a column of air of crosssectional area 1 square inch extending from sea level up to in nity The atmospheric pressure p at an altitude h above sea level is the weight of the air in this column above the altitude h 393 Johann Heinrich Lambert 17281777 was a Swiss German astronomer mathematician physicist and man of learning He was mainly self educated and published works on the orbits of comets the theory of light and the construction of maps The Lambert equalarea projection is well known to all cartographers He is remembered among mathematicians for having given the rst proof that 71 is irrational pq 2 29quot DIFFER P 39EUUHi39I 1 H5 Aseumiing that the density ef the air pzrepertiehel ter the pressure shew that p sieiiis iee ihe diff7E39fEn tiai eqgiieiiieii e en E e and evhiftein the feemiuie p Iequotquot39i where Pg the etmeepLheiie pre5e39ure at sea ieeeL Assume that the raise at erhieh a heel hedy eeele is iprep e39rtienai tea the di ereeeAe in temperetiuire hetiweiee it and its euneundinge Newirewe iiew elf eee ieg39 0 39hedy ie heated the i11iEi C eiri pleeed in air at After 1 hour its Iempxereiime is IIIew mueh edditienel iimei is requireii fer it he eeel tea A bed ef imivznewrn temperature placed in e freezer w39hieh ie kept at e eienetem 1iEFl391P l39ampIl1iIE ef EFF Mter 15 mimitee the temperature ef the hardy ie 3 fF efrd after b minutes 15quotFi What was ti1e ini eli tempereture erf the Abedy Selve this problem miereiy fI39ITiIl39Ci1 Ig eheet it 31111 eiee by ee Iainig e euitelle di ieriemieil evquetiiem A peat ef eerret eindgarlic seep e ee iing in air et WC ewes iIiitieli3r beiiiihg Eli e anid eeeled 20 during the hrei 3 U miinm ee Hew much will it eeel Aeuiring the next 30 imiiinetes Fer eh wees reeeene the iieeee 1ingfeeAmi ef e eertein eemner ie kept very eeeil et ea eenetenji iempiereti1re of SEC if E While gtdeing en eu tepey39 em eheh memeing em a muird er iquoti39EtilTL the eemner himself 0 killed heed the viveiiimie heel is eteieni p ii em the eerehei s eesieteini diisieeeers hie chief39s body and riciie its 39ie39rn1pere39tiire he he 23 quotCi and et neeri the hedg e temp ereiuirei is drwn te i 5 C Aeeumihg the eerecner heel e nermeJli teingperetzure ef 3iquot C C 939B F when he wee eilivei when was he in L1 rde reel The 139edieieerhen in li i39Ei g weed decays at the grate ef dieir1tegreiie ni5 per minute quotdLpmi per grerri f eenmtined eerheni Ueirtg 3eerie Fee ihe heiilifie eif reLiieeeirihen eeiimeiie the ege eff eeeh ef the ifeilewiiing epeeimeee elieeievere1 ereheeeiegiei5 end ts ii d f ff Ieidieeet39iquoteit in 1950 e e pieee if e eihlteir Ileg fremi the temb erf King Tetenkhemen 1014dpmi h e pieee emf ye beam ref e heuee hei1i in Bahgrl en duiririg the reign elf Memmuriehi 952 dpm we dung ef e giant eiveih f lJ 39d 6 feet 4 iinehee lliWld39 F the surface ef the geeLr id ineieieu Gygpasiuirei Cave in Neeedei 1 dpm M 3 herdwed eteti epereritihrewer feeed ih Leieeeirii iReeki Shelter in Nevada iipm H Hiew tein himeeif eLppied this zmte ten eiimeie Efhe Ierrnpe re Iere ef e redwhei iren hei1 lime was imewh ehewil the laws ef heet tie ef eir ai that time ihai hie reeuit was en lj3 39 e ieugih eppmIimeIiven but it waeeerrIeiAIe391Iy better than 39neEhing 395 The idea fer ihie peeiljilem P eiee te James F Huriejm quot39Ah Appiivee1iien ef Nnewtei39nquotei JLnII ef vIEfeiIing The imerhemeiriee Teeeher weli 6 i 394 i pp M1E THE NATURE OF DIFFERENTIAL EQUATIONS SEPARABLE EQUATIONS 5 FALLING BODIES AND OTHER MOTION PROBLEMS In this section we study the dynamical problem of determining the motion of a particle along a given path under the action of given forces We consider only two simple cases a vertical path in which the particle is falling either freely under the in uence of gravity alone or with air resistancequot taken into account and a circular path typi ed by the motion of the bob of a pendulum Free fall The problem of a freely falling body was discussed in Section 1 and we arrived at the differential equation L5 dtz 1 for this motion where y is the distance down to the body from some xed height One integration yields the velocity d v gtc1 Since the constant c is clearly the value of u when t 0 it is the initial velocity v0 and 2 becomes v v dy t 3 dt 8 0 On integrating again we get 1 2 U0tC2 The constant C2 is the value of y when 1 0 or the initial position yo so we nally have 1 Y 3 Egtz i 7101 Y0 as the general solution of 1 If the body falls from rest starting at y 0 so that U0 yo 0 then 3 and 4 reduce to 1 2 vgt and yzigt On eliminating t we have the useful equation vZ9 5 30 iiiIFFEiFlEHTTIAL E UAquot1T HE far the VKCIDEiitff attained in terms uf tiiie di5IIanue falieni This result can iS D bi i iibiai Eli fmm Aprimipiie of wn5Eruatiimni if enAergyi which can be stated in thei form kinetic Bniei1r1gy pmcntiail merigy a rcmstanti Since Mr handy falls fmm rest starting at U the fact its gain in kiirmtiic nergjr Ezquals iits I055 in PiEI1quotZi i im rigy give quot1 mug and 5 failoiws at mnce Remrded we assume that 3i139 ELKEFIE a iriesisiing f gf Ev pimpmti anai t the quotI squotEi EJEiIf mi DIUIT ifaiiiinig huciyi than the idii entiiai Equatinin Di the mtium is N g o 3 W iwham c SEE Eq1 IiiI39iD 1BJI if idy dt is riepiiaimdi by U ibiemmesi dv P p rzu 7 On mpariating vairiaiiaa and intiagranting we get divquot Hui it an 1 i EEingirg 4 wji ch 1r E1 i C19 Tquot initial CEl diti fi v 2 when I 0 gives 5 g so H i i 0 P Sinima if is pnsitivz u p gift as E H This limiiting value Elf U p called the rgrmiiirmi ireiaciryi If we wiish Awe can Maw r piamp4te v by dyfdir in Si and pmfsi nn i1 thE1 ii1it3gr Eiit3in ED I ui y as 3 functi ni f The m lihl Inf 3 jpen ulumi Caizzrnisiider a pafnduium mni sii5itini of 3 bub miquot mass m at the and of La rml Elf niagligible mass and iiangth If itihE bah is THE NATURE OF DIFFERENTIAL EQUATIONS SEPARABLE EQUATIONS 31 FIGURE 7 pulled to one side through an angle or and released Fig 7 then by the principle of conservation of energy we have mv 39 mga cos 6 a cos oz 9 Since 5 a6 and v dsdt ad6dt this equation gives 1 2 d6 2 ga d t gacos 9 cos C1 10 and on solving for dt and taking into account the fact that 9 decreases as t increases for small t we get a d9 m 2g cos 6 cos a If T is the period that is the time required for one complete oscillation then I if d9 4 2gcos6 cosoz d9 T4 3 u 2g 0 cos 9 cosa The value of T in this formula clearly depends on oz which is the reason why pendulum clocks vary in their rate of keeping time as the bob swings through a larger or smaller angle Formula 11 for the period can be expressed more satisfactorily as follows Since by one of the half angle OI 396 This dependence of the period on the amplitude of the swing is what is meant by the circular error of pendulum clocks 32 DIFFERENTIAL E UATI 139 IS fnrmtulas of ttrignntmmattryt we havae 8 cuts 8 2 1 Estinzif and 1 t a 2050 1 Esrnz za we can twritte pCr 4 E J t 79 T 39 2 Q J vsinl Eta aw em V 1 V d N quot p L E A E J P V k 51111 2 12 WE aw thangs the vwitattailt fmm B to Q by puttittg sin 939 03 sin 50 ttlmatt intreaset5 from 0 t0 3391 f2 as H inC rBB1SE5 from U tn ac and g 9 Ecusid kcos d Dquot 2k ms d zvkt sit12H f2 M n5H2 V1 Main This EHBMES us ta twrite 12 in tha farm a 39 E d Itt T quot r 3 A 0 AH n kl S1IT1239 is a fILli IE39I7 i lFI Bf It and ca llted the effiptic tfnttegtrai 0f the rm ktfrtdi 17 Th e iptiu fnIEgr It39 0f the 39i E39C d kii g 3995 Em M f 1 U W hEI39 0 o t at am Po d r arises in mnttetsttiuzrttrm with tha problem of nding the circuxmfemtnce of an 3llitpsE see Pmblem 9 Theuse elliptic integtrals E li be Equotu h1 tE d in tetquot139Its f ElBmsmtar y fltl tl illlti l Since th 3V men quite fr qllIe t ly in apptlicattinns tn phjysicst and Engineering fh if values as nutmtaritzal func rns f k and T am often given in matfheemattjtal tablet It is t39ust0mary in the case f elliptic inatcgra39s to vintat mrdinary 1Isag by atlm ng the same ett1r tn 3jpp I39 as th ttpp I liItttt attusti as the dtummy vati sabte rif itttcgtatitm THE NATURE or DIFFERENTIAL EQUATIONS SEPARABLE EQUATIONS 33 Our discussion of the pendulum problem up to this point has focused on the rst order equation 10 For some purposes it is more convenient to deal with the second order equation obtained by differentiating 10 with respect to t d29 a ampl3 g sin 6 14 If we now recall that sin 9 is approximately equal to 6 for small values of 6 then 14 becomes approximately d26 5 a dtz 0 15 It will be seen later in Section 11 that the general solution of the important second order equation dzy k2 0 dxz y is y c sin kx C2 cos kx 6 c sin gt czcos gt 16 a a The requirement that 6 at and d6dt 0 when t 0 implies that cl 0 and C2 or so 16 reduces to 9 acos gt 17 a The period of this approximate solution of 14 is 23 It is interesting to note that this is precisely the value of T obtained from 13 when k O which is approximately true when the pendulum oscillates through very small angles so 15 yields PROBLEMS 1 If the air resistance acting on a falling body of mass m exerts a retarding force proportional to the square of the velocity then equation 7 becomes dv 2 1 g cv where c km If v 0 when t O nd v as a function of t What is the terminal velocity in this case 8 10 11 13 35 THE NATURE OF DIFFERENTIAL EQUATIONS SEPARABLE EQUATIONS Inside the earth the force of gravity is proportional to the distance from the center If a hole is drilled through the earth from pole to pole and a rock is dropped into the hole with what velocity will it reach the center a Show that the length of the part of the ellipse x2a2 yzb2 1 a gt b that lies in the rst quadrant is a a2 e2x2 39quotquotr 7 dx 0 a X where e is the eccentricity b Use the change of variable x a sin 4 to transform the integral in a into aft V1 6392 sin2 d aEe Jr2 so that the complete circumference of the ellipse is 4aE e fl 2 Show that the length of one arch of y sinx is 2i E T71112 Show that the total length of the lemniscate r2 a2 cos 29 is 4aFZ14 Given the cylinder and sphere whose equations in cylindrical coordinates are r a sin 6 and r2 22 b2 with a S b show that a The area of the part of the cylinder that lies inside the sphere is 4abEab1r2 b The area of the part of the sphere that lies inside the cylinder is 2b27r 2EabJr2 Establish the following evaluations of de nite integrals in terms of elliptic integrals J1 2 dx 3 Kn mg p2FV12Jr2 hint putx 112 y then cosy cosz 45 b J2 Vcoisxdx 2 E17212 FT2i2 hintz put cosx cos2 41gt c Ln2 V1 4sin2xdx E4in2 hintz putx If2 lt13 6 THE BRACHISTOCHRONE FERMAT AND THE BERNOULLIS Imagine that a point A is joined by a straight wire to a lower point B in the same vertical plane Fig 8 and that a bead is allowed to slide without friction down the wire from A to B We can also consider the case in which the wire is bent into an arc of a circle so that the motion of the bead is the same as that of the descending bob of a pendulum Which descent takes the least time that along the straight path or that along THE NATURE OF DIFFERENTIAL EQUATIONS SEPARABLE EQUATIONS 37 A I a I I I 39 X I I I I I I I U2 I I I I I C FIGURE 9 methods of elementary calculus we nd that x c x v1a2 x2 v2b2 c x2 or sin al sin tag U1 U2 This is Snell s law of refraction which was originally discovered ex perimentally in the less illuminating form sin avsin ozz a constant 19 Willebrord Snell 15911626 was a Dutch astronomer and mathematician At the age of twentytwo he succeeded his father as professor of mathematics at Leiden His fame rests mainly on his discovery in 1621 of the law of refraction which played a signi cant role in the development of both calculus and the wave theory of light The asauaaaaptamn that light travwalcs fmm an paint ta aaaathaar aalang the path reaqu irianga the 5hUI1iESIt tiima is caa lad FearmaIa 5 prinaipfa af feast time Thi7a prinaiplea mm manly gapmav39iaadea a rational 7baaii5 fara Sane fa laaquotwa Rbm Cm aalsa be applied to nd the path af a rayaf tfhmugh a amadiaam aaf avariabla daaaaaiyty where in anazral Iight will travel along aurvea jwnsrtaad atquot staraaiaghyt linea In Fig h we laaava a satrati avaa aptiaalj madium In the aindiavildiaala layaaara atht valacirtgr of alight aanastaantgj but the aaewaaaya damaaaasaa fI 39lTTl each layer ata the ant Abalmw 9 Ih393 deraceiadling ray of passes L ilayer tan layer it is FEWEEIEda rare and mm toward ma verticaka an whana Snall s law is aappiliiad ta a baanaadariaazs beatweaan the layers we uobtairn ainTar siiaa52 ain atg sina4 U1 U3 If we E El allow athaasa llayers ta gmw thinaar and amrae nuamaraus than in the limit the velocity nf lighat dasrraaaas cantianvaauaaalyw as the ray daacanadaa and awe aarna1aude thaat aim at a aana39taaaat v This situation is ia dicatad in 79 and is aapapraaimataa ay what happana ta a rafraa of aumligaht falling an ma aartah as it slaws in dasaandainag thraagVha atmaaaphaea of maraaaaitng daaasiaty Rartuminga aw ta EarAnaAuaMi s prablemi we i n Imdure a Ea ir di39m13 tE axystaaam as in iFaiag aad imagine that tha ataadi like the ray af light is M I gt p 10 THE NATURE or DIFFERENTIAL EQUATIONS SEPARABLE EQUATIONS 39 I capable of selecting the path down which it will slide from A to B in the shortest possible time The argument given above yields sin a U a constant 1 By the principle of conservation of energy the velocity attained by the bead at a given level is determined solely by its loss of potential energy in reaching that level and not at all by the path that brought it there As in the preceding section this gives 1 0N0 2 From the geometry of the situation we also have 1 1 1 S1nacOS Sec p On combining equations 1 2 and 3 obtained from optics mechanics and calculus we get y1 y392 C 4 as the differential equation of the brachistochrone We now complete our discussion and discover what curve the brachistochrone actually is by solving 4 When y is replaced by dydx and the variables are separated 4 becomes dx y 12 dy 5 C At this point we introduce a new variable qb by putting C i yl2 tan 4 6 so that y c sin2 1 dy 2c sin 1 cos 1 dqb and dic tan qb dy 2c sinz cp dqb c1 cos 21 dqb Integration now yields C x E2 s1n2ltb c Our curve is to pass through the origin so by 6 we have x y O IFFEFE39Iu TIA1 EuU TI H5 Wham U and mnstaquently 1 z Ttmrs x 2 at 5 sin 2gb T and I Ft y 2 train 1 cts2 8 N If we n tw put a t c fl FI39ad 9 M then G and B 39becnmE x aw sin 6 anti at a l cos IE9 Thazse are the Etat1datquotd phramtettarittt Eqtttitli s f the Cy l i shmswtrt in 0 11 p is gr rer t d by a pint cm the E2il39Cuf 39fET396 E Elf at scircllc hf radifus 3 mlltingg lflg the xuattis We fl t p there is a s ingh value hf a that makes ha rrst atquotth hf this ctty ctmttd pass thmtth tha paint 32 in W for his fi1iHUW Ed tn increase from 0 the wt than tlizhe arch rin atEs tweepst G vquotEl thteh rst quadrant of 0 plane and clearly p SES tthmugh H tf r a sittte 5 uitabl ch sen value of hi Same hf the g E0 metric pr tpertites at the tyclhid are pethtaps familiar In the readeI fmttt elametnttatry calculus F tt restxtampttm thh length hf U113 arch is 4 tgitmas the rdiatmeterr Elf the gETl lEI 39 ti1 lg tcirr2Ea and the htre tutnzdlar ohm arch is 3 timhs 1112 area at V ri1rwJltc This ramarkahle mirwe has many mtther inttetestting pfCtEI tiB Ei bUth geometric anzdt phystiacalt and same of thtse are described in the prhh tms htl0wtt We hlmpet thatt the necessary details have ht io1hscure d the twttndarful imagitthaTttit39e qI1E3IjtirES in Bttm uMiquotS Vb EhiS IDEhFnE plCtbtl f and his st utihn Elf it f tlf this whale ttrttactturc 0f th ught is a twcnrk of i 393939Efv1iBCI l tart Infquot 3 varyt high nrdtett In additihn t its iIlTtIquotil Iz5i i t ir tt the K f ChiE5tt 39EhtID E prnttutzm has a Harper signi cancev it was the hittttutricalt S tttttlet at the chtcutus hf Uj 7Ff i fv 39nS 3 phwerful branch of t lF5iS that itt mhdern 39IZi139l39lE5 thtas p nettated deeply izntu th hidden tsimiplitcrittiet at the J ills FIISUHE ll THE NATURE OF DIFFERENTIAL EQUATIONS SEPARABLE EQUATIONS 41 heart of the physical world We shall discuss this subject in Chapter 12 and develop a general method for obtaining equation 4 that is applicable to a wide variety of similar problems NOTE ON FERMAT Pierre de Fermat 16011665 was perhaps the greatest mathematician of the seventeenth century but his in uence was limited by his lack of interest in publishing his discoveries which are known mainly from letters to friends and marginal notes in the books he read By profession he was a jurist and the king s parliamentary counselor in the French provincial town of Toulouse However his hobby and private passion was mathematics In 1629 he invented analytic geometry but most of the credit went to Descartes who hurried into print with his own similar ideas in 1637 At this time 13 years before Newton was born Fermat also discovered a method for drawing tangents to curves and nding maxima and minima which amounted to the elements of differential calculus Newton acknowledged in a letter that became known only in 1934 that some of his own early ideas on this subject came directly from Fermat In a series of letters written in 1654 Fermat and Pascal jointly developed the fundamental concepts of the theory of probability His discovery in 1657 of the principle of least time and its connection with the refraction of light was the rst step ever taken in the direction of a coherent theory of optics It was in the theory of numbers however that Fermat s genius shone most brilliantly for it is doubtful whether his insight into the properties of the familiar but mysterious positive integers has ever been equaled We mention a few of his many discoveries in this eld 39 1 Fermat s two squares theorem Every prime number of the form 4n 1 can be written as the sum of two squares in one and only one way 2 Fermat s theorem If p is any prime number and n is any positive integer then p divides n n 3 Fermat s last theorem If n gt 2 then xquot yquot 2quot cannot be satis ed by any positive integers x y 2 He wrote this last statement in the margin of one of his books in connection with a passage dealing with the fact that x2 y2 22 has many integer solutions He then added the tantalizing remark I have found a truly wonderful proof which this margin is too narrow to contain Unfortunately no proof has ever been discovered by anyone else and Fermat s last theorem remains to this day one of the most ba iing unsolved problems of mathematics Finding a proof would confer instant immortality on the nder but the ambitious student should be warned that many able mathematicians and some great ones have tried in vain for hundreds of years NOTE ON THE BERNOULLI FAMILY Most people are aware that Johann Sebastian Bach was one of the greatest composers of all time However it is less well known that his proli c family was so consistently talented in this direction that several dozen Bachs were eminent musicians from the sixteenth to the lErEHE139t i IsL seuattess nineteenth eenturies In fact there w ere parts ef Germattjt where the sherry weird hash trteant a l Eltll5 LiE39ii1It What the Bash sen was te m39asie the Eermeuliis were its n1at39hetrIaitius arid seierlee In three gerteratierns this remarkalle Swiss family pressed eight mathematieiaasthree efi tgherrt euitrsitahtitirtg ishve in turn had a swarm est destertdants whe clsistirrguishtettl themseflves in many welds Jiarttes Betrne ulli 1 54 1r39G57 stiudieti th ee legy at the insistenee elf his father but lahartdenetl it as seen as pessihle is faster sf his 1ev39e fer seienee He taught hirnselt the new ealeulusr if Newtetrl and Leihnia and was tpretesser ef matihematies at Basel item 168 urttil his rleath He err e te en irrhnlite series studtiteel many speeial euraes instenited pelar eeerdinates and i l1 quotrEJTtiEeCi the Eerrneulli numbers that ippeeI7 in 29 pmser series IE 3E39pa39I39lEi l39I ef the tunstien iIan as In his 0 Ara Ceejseseedi he termulatecl the hasie priuei ple il l the theert ef pire39h a hilitjy lanewr1 as Btetrneeiliis theerem er the iew of ierge numbers if the spree39bahility elf a certain event o p artti if H indepehdent trials are made witih E st1eeesses them his a p as pB 3 mt At rst sight this srtatemeant may seem ts he a tris39ali39t htrt hetteath its serftaee lies a tranglted thiieiset ef philesephrieal arid tnatihernatieal prehlemst that have heart a seuree elf een39lrev39ers1t them EEI ULiiii E time tea the present day Jlames7quots i iJl lg El39 hrthet39 Jehh Eernehllil I39 166 tiTquotT also tried a talse start in his eareer he sttidjyirtg medieihe arid staking a dDCI 1quoti5 degree at Easel in ti e eith a thesis en rrtt1seIe eentraetierL Hmseser he alse hecame faseinated hjt eaI eLtlt1s qttieltily mastered it and agppli e it ts manly prehlerrts in geeree39tr3r tli ereatial equattietast and mechanics in he was tappeirited pre iesser lei mathersaties and physics at GI39I39li gE in i39l39e1iartd and en I imE5iS sleath he sueeee ecl his hr etChe r its the prefessership at Basel lite Bampt39 D39liii l brtherst setmet irnes wrerhee en the same prehlerrts whieh was unfertlmate in View ef their jealous and teaehgt dispesiitiertsr Ott J39CJ3Si n the frietiee hetween them hated it isle a hittzer aruti ahteisve ptuhlie feiue as it tjliti ever the hraehisteehrene prehlerttt It 15 Jehrt I39Dfp SE i the pr e39b1em as a c halLlet1ge he the mathematiieiarts sf Ettrepe It rl a di great iriterest and was selvteti by Newtett arid Leibniz well as he the has E etrt1euliis Jlehrlis seilutien whitzhr we have steam was the zmsehs ellegartt while Iames39s theugh rather slums arul laherieus wars metre general This situatiert started an aerimenieus quarrel that tzlragged en fer SEquott eTtquot i 3quotears arid was eftett eeneltteted in reugh iattguage mere 5lilit dr te a street brawl than a seiertilie gdisettssieri Jehrn appears the have heett the mere E39 iFI i i iELf TJ ef the twe fer 39rnLwh later in a fit eif jealeus rage he tihreat his sewn Sii1quottUl139t Let the hettse fer winrnlitrg a prize frem the Freneh Aead emjt that he eeseted fer himtself This ssh Dartiel Bertteuili 1fT U i32 sitrudieei rhedisirne like his tathrer and teelt a degree with a thesis en the aetien ef the lurtgsg anti trike his father he seer gave way te his inherit talertt and heearrte ta prefesser st rr1athert1atiss at 6 Pettelrstiburg In he rettrrrned te Easel arid was 5 llCCIES5tl quotIa E13 1r39etesse r sf he39tan3rt anaterr1t artcl 1hysies He were it prises frern the Frerteh Aeadvemy ihelmlirtg the erte that ihiuriat ed his hither and ever the years pt139hlistl1etdr marry tfterks mt hrsies preh ahility ealeultls and di er erttial equatiehs lrt his fameLIs heel diiquot i39 it iiiEt he diset1ssed ttiti I39tteehanim and gave the Aetatllest treatrrtettt at the ltirtetie theaters let gasest He eertsidered hay many as ihave heart the tits ge11391i39tli11E mathzematieafl physieisttr THE NATURE OF DIFFERENTIAL EQUATIONS SEPARABLE EQUATIONS V PROBLEMS 1 It is stated in the text that the length of one arch of the cycloid 9 is 4 times the diameter of the generating circle Wren s theorem Prove this 2 It is stated in the text that the area under one arch of the cycloid 9 is 3 times the area of the generating circle Torricelli s theorem Prove this 3 Obtain equations 9 for the cycloid by direct integration from the integrated form of equation 5 x I Vc y dy39 by starting with the algebraic substitution u2 yc y and continuing with a natural trigonometric substitution 4 Consider a wire bent into the shape of the cycloid 9 and invert it as in Fig 10 If a bead is released at the origin and slides down the wire without friction show that Jr ag is the time it takes to reach the point Jra2a at the bottom 5 Show that the number an37 in Problem 4 is also the time the bead takes to slide to the bottom from any intermediate point so that the bead will reach the bottom in the same time no matter where it is released This is known as the tautochrone property of the cycloid from the Greek tauto the same chronos time 6 At sunset a man is standing at the base of a domeshaped hill where it faces the setting sun He throws a rock straight up in such a manner that the highest 20 Christopher Wren 16321723 the greatest of English architects was an astronomer and mathematician in fact Savilian Professor of Astronomy at Oxford before the Great Fire of London in 1666 gave him his opportunity to build St Paul s Cathedral as well as dozens of smaller churches throughout the city V 2 Evangelista Torricelli 16081647 was an Italian physicist and mathematician and a disciple of Galileo whom he served as secretary In addition to discovering and proving the theorem stated above be advanced the rst correct ideas which were narrowly missed by Galileo about atmospheric pressure and the nature of vacuums and invented the barometer as an application of his theories See James B Conant Science and Common Sense Yale University Press New Haven 1951 pp 6371 The geometric theorems of Wren and Torricelli stated in Problems 1 and 2 are straightforward calculus exercises for us It is interesting to consider how they might have been discovered and proved at a time when the powerful methods of calculus did not exist 22 The tautochrone property of the cyloid was discovered by the great Dutch scientist Christiaan Huygens 16291695 He published it in 1673 in his treatise on the theory of pendulum clocks and it was wellknown to all European mathematicians at the end of the seventeenth century When John Bernoulli published his discovery of the brachistochrone in 1696 he expressed himself in the following exuberant language in Latin of course With justice we admire Huygens because he rst discovered that a heavy particle falls down along a common cycloid in the same time no matter from what point on the cycloid it begins its motion But you will be petri ed with astonishment when I say that precisely this cycloid the tautochrone of Huygens is our required brachistochrone p DI FFEREETFIA EDIJ tTi U39H3 peitrtt teaches is level witth the tep ef itih hilt J t the reelt rises its Ashedew mteuest esp the surface ef the hill at 3 eeesttsnt speed Shew that the pro le ef the hit st eye ieitd CHAPTER 1 1 9 It btegaxn tee snew en a eestein metrnitntgi sand the sneer eenttinued te fail stetsdiiy thtreuigi1evut the day M tieen fa stsewpitew started tee etesr a read at e ee nsts1tti rate in terms ei quotthe seilittme ef snew temevetJt fpE it heitm S snewplew cleared 2 miles by V PJHL anct 1 mere mile by 4 IFM Whtetn die it start sinewing A met hbsl whese rediusi was earieginellly iinehi l feund te theme e FadituE ei ineth after 1 menttL tss umting that it e39srspettstes at a rate prertienst te its isurfeeet nd the radius as P z ftmetitm ef time After how many metres miiths wiillll it tgltissppeert sItege th et A itittf ermttsins 391EIt gslttens f pure wstezr Beginning at v it D brine t 39 tiEii i g 1 pound seWgsIietn Hews in st the tste ef 1 gel1 nfminu te and thes mit t1tte whieh is kept L 5iIquotIIiftD39I T39I39l by stirI ittg t WE ettt at the serne rates Z witlfi there be 550 petmtls of diS5DLlttEd ssit in the stems m large quottents ee n t sirts ICE egelliliems ef brine in which 0m petmtis ef Assist are distsehtetd Btetginning at time I fl pu re water f ews in at the rate net 3 gallonsI t1itnute end the ntittttJtquot wlirieh is kept entiferm b stiisringjt ews net at the rate elf 2 geellertsfminutue Hew letig will it take to redttee the 31t39l39J 39i1lJl fIIi of salt is the tank to p 39pDu39 dS smeetth teetbelt hetvting the shape ef en eJ1ipsetitd irnethes ting and 6 inehes 39IhiCl is l3r39ing euttdeers in s r it1sItftv39trtt Find the paths elem wttitzh wstezt will t39quot IZt dewn its sides If e is s spesitiv e eenstsnt ends is is s pesiiittise paremequottes the I2 E I Q HE E 2 39 E P the reqtjatiQTI ef the femilt eat all eIiipses e T 42 and 39hperb ies is z ti wicth feet st the peitntst es U sSl1ew thstti this quotfamilyquot ef t39 f ti39vtIi39 castes is seifert39hegettsij see Prtemem mJ A eeetcItitng ten Terrieeiffs iititIti wetter in en eperl tEii wills ew eut t hre39ug11 a smell hete in the bettem wi tTh the speed it wetuld acquires in fetliriig freely tresn the waiterquot lestet tee the hie A t1 emistphetiesi bewl ef trasjitus R is tinitistlgr fullt ef wat er and is small eireL as heie est rtsdtittsi r 4 PE i dii in tthe be39ttem at time t 4 Hew Eeng wilt the how telee to emjptfy itsef 1 The eitepsr39tdrsjt er sneiient wetter eieek was a bowl tiem ssitieth wetter west sllewed te escape 39thtetiigi391 is smsi i hole in the bett3m It wes eftent used in Greek end Rerrtsn C tltts tee time the speeches ef istwjgwers in order the keep them items ttsiking tee fEtUEh tFimt the shapes it shettiti hasvie p the water itesei is ten fs7ll all e enrtstst39nt rate Twe espew tisnits witth itdentiieisl smstl heiress in the bnttern tlreirt in the time Dee is is c3rI inrIetr with s vertieslt ssis atttI the ether p is sense with trerteit THE NATURE or DIFFERENTIAL EQUATIONS SEPARABLE EQUATIONS 45 down If they have equal bases and the height of the cylinder is h what is the height of the cone 10 A cylindrical can partly lled with water is rotated about its axis with constant angular velocity u Show that the surface of the water assumes the shape of a paraboloid of revolution Hint The centripetal force acting on a particle of water of mass m at the free surface is mxwziwhere x is its distance from the axis and this is the resultant of the downward gravitational force mg and the normal reaction force R due to othernearby particles of water 11 Consider a bead at the highest point of a circle in a vertical plane and let that point be joined to any lower point on the circle by a straight wire If the bead slides down the wire without friction show that it will reach the circle in the same time regardless of the position of the lower point 12 A chain 4 feet long starts with 1 foot hanging over the edge of a table Neglect friction and nd the time required for the chain to slide off the table 13 Experience tells us that a man holding one end of a rope wound around a A wooden post can restrain with a small force a much greater force at the other end Quantitatively is is not di icult to see that if T and T AT are the tensions in the rope at angles 6 and 6 A9 in Fig 12 then a normal force of approximately TAB is exerted by the rope on the post in the region between 6 and 6 A6 It follows from this that if J is the coe icient of friction between the rope and the post then AT is approximately uT A9 Use this statement to formulate the differential equation relating T and 6 and solve this equation to nd T as a function of 9 u and the force 7 exerted by the man TAT FIGURE 12 46 1amp1 15 16 IIL7 H H L E U t39Tl iN5 p teed L is ettppeetted by e tapetre J ifr t atr Ci llIi wheee meetetriel hes dwet15it3r e If the red itue Of the I 1 ef the C Utm erg nd the radius r at e dtietetnee J heiew the temp the areas Elf the thm39izuentel eteee eectitltte ere pureetrtiehatl ten the tetet leads they heer The Pteeident end the Primet Mtiniett et erdere ee ee end FESEi39vE eepe ef equal etempeteture et the time The IP1re5iiden39t edde tee entail Et1 I1t 39atJnt of etteel erfeam imtmetdtteetetyt hut ees Jrt l dfFi k his eet tete until Ill mthuttes tatert The Prime Mineieter wetits 110 39til1L 5E 5 and then ads the eeme ezmesutlt hf eeel cream and hegine tn dhrritrtil2 Whe drinks the htrtter ee ee 0 destroyer ie hunttitng e e39uhmerine in ex dense feg feg ltfte the e tmmmentt diieetesee the suhtmerine en the SttJ I394l39ft1 E 3 n tee away erttl atetly d eeeende The epeedt evf the de539trnyer is twiee the ef the euhmerinre and it ltitewh that the IIEIEET will at enee dive tenet depart at full speed in e etretghtt weree hf tJ k39tD W dtireetiieht What path ehetttd the des39tre1rer39 felhew to he certain ef pasetinsg ditquotEEHt 39t39ElT the eruJhmerine39 quotHint tEe39te39htish e peter eeetdinete system wtith the hrigitt at the pint whtetve the submarine was sighted Fteuzr quothuge sit at the teesmerre ef e square telhliet ef side e At the same instant they ell begin tn wetkt with the seme epeettI eeeh mev39ing eteedittre tewerd the httg en right If e peter eeterd391 ne te 53aet em is eetethliehed en the tehle 0 the eriin ext the ttEI ttEt39 and the peter ettie ellemg e diegeheL nd the path ef the bug thett srtezrte en peter attie end the total ietettee it twattes beteee all httge meet at the eemerz CHAPTER 2 FIRST ORDER EQUATIONS 7 HOMOGENEOUS EQUATIONS Generally speaking it is very difficult to solve rst order differential equations Even the apparently simple equation d 3 ifx2y cannot be solved in general in the sense that no formulas exist for obtaining its solution in all cases On the other hand there are certain standard types of rst order equations for which routine methods of solution are available In this chapter we shall brie y discuss a few of the types that have many applications Since our main purpose is to acquire technical facility we shall completely disregard questions of continuity differentiability the possible vanishing of divisors and so on The relevant problems of a purely mathematical nature will be dealt with later when some of the necessary background has been developed The simplest of the standard types is that in which the variables are separable f gltxgthlty 47 43 m R39EHT L quot1 U nT139I39H 5v As we kmw 1 swlve EMS we haw Grim to write it in the separated fmm dyHzy gfxj and integrate We Ahava seen many examgple mf this pmcedzura in the pre cedAing cha1Jtsr h nezt Havel of cT mp3Eexi t r is t1h h0mngeneV u5 eqVuati0n A fullcti n I y is 39EEl JEd mmagtetnawus of gdggree M if f W for all EL1it3 lIJ ljf resAtrivcIte d 3 My amid H py maacans that if 3 and y are fEl 39CEd Aby 0 and y I factmrs nut mi the re5uLlting fl1IlG tTJl7ID L d the mmainingA factor is the miginall fur1mtiuI r1 Thus xi Axyg I 9 and 55111 1fy are Vham gemeiarus Hf degrees 2 and U The disi erentiaflj equ n p NVt1ydy J is said ta be J10mag1e ne us M anal are hnm0gme0u5 39f L1I 1CIiD s of the same deree equativr 1 can then ha writ1tan in the fmm 35 y m W hEl E fry p6 Mx yANx y is clearly hnmAUg1emmu5 f degree The pr cadure far su ivimg 1 rests an the fact ihat it can I ways be changed irmtm an equatinn separablE quotuquot I39i blE3f5 m e an s 0f the substijtminn p yf reg uquotdles5V If the of the fAunvtin fx5y To see this We Hate lzhat the Iw1atwViVm1 permits US tn set E 1quot 1 and bmin Vf39M f1s39N9 f1E Then since 3 2 and dz 32 H5 2 equation bEc me5 p k 3 f1p and the variaables can be 5EpaI39E11E dE dz H f1z 2 J 39 We n w m mg3late 5l uVtimn quothyr integratir1g and rEp1aainVg 3 Why FIRST ORDER EQUATIONS 49 Example 1 Solve x y dx x y dy 0 We begin by writing the equation in the form suggested by the above discussion Since the function on the right is clearly homogeneous of degree 0 we know that it can be expressed as a function of z yx This is easily accomplished by dividing numerator and denominator by x gy 1 yx 1 2 dx 1 yx 1 z We next introduce equation 2 and separate the variables which gives 1 2 dz gic 1 22 x 39 On integration this yields 1 tan 2 log1 22 logx c and when 2 is replaced by yx we obtain tanquot log x2 y2 c as the desired solution PROBLEMS 1 Verify that the following equations are homogeneous and solve them ax22y2dxxydy0 f xydxxydy0 b xzy 3xy 2y2 0 g xy 2x 3y C xzy 3x2 y tan 3yC xy h xy Vxz yz d d xsinXlysinXx i xzy y22xy xdx x 6 xy y 2xequot J x3 y3dx xyzdy 0 2 Use rectangular coordinates to nd the orthogonal trajectories of the family of all circles tangent to the yaxis at the origin 3 Show that the substitution 2 ax by cchanges y fax by c into an equation with separable variables and apply this method to solve the following equations a y x y2 b y sin2x y 1 FIRST ORDER EQUATIONS 51 8 EXACT EQUATIONS If we start with a family of curves fxy c then its differential equation can be written in the form df 0 or 3f W39 axdx aydy O For example the family x2y3 c has 2xy3 dx 3x2y2 dy 0 as its differential equation Suppose we turn this situation around and begin with the differential equation Mxy dx Nxy dy O 1 If there happens to exist a function f x y such that af af ax M and ay N 2 then 1 can be written in the form W39 i d d ax xay y 0 or if 0 and its general solution is fJc In this case the expression M dx N dy is said to be an exact differential and 1 is called an exact differential equation It is sometimes possible to determine exactness and nd the function f by mere inspection Thus the left sides of 1 ydxxdyO and dx dy0 are recognizable as the differentials of xy and x y respectively so the general solutions of these equations are xy cand x y c In all but the simplest cases however this technique of solution by insight is clearly impractical What is needed is a test for exactness and a method for nding the function f We develop this test and method as follows Suppose that 1 is exact so that there exists a function f satisfying equations 2 We know from elementary calculus that the mixed second 52 D139FF1aHJEhrTiaT EDUATEDrN5 partial di er ivatives f f are equaali 331 E ii 3 SJ SI d d d V p J 4 y 81 so 4 is a me1ressary cnnditiann fear the exa ntn 5s nif 1 We shall ja mvE that it is lS i 5u Fmient by sh wi39ng Iihajlti p enabl ss us tn cnsItr uEI E4 393 This 39iEIJIJS funttimn f that equatians 2 We begin by integrati n rim rs azaf aquamisans 2 with TEEp t to Jr IMiran 6 The mntant inf integratin quot murriing ham is an arbiitrary fun ttij n Bf y since N must diappear under differentiatimVn Vwim respect to 1 This reduces OUT problem to that Inf nding a Euhnction gfy 39with the prv f perty that 7215 given by 5 saIi5 e5 the seczurnd nf equatTmn5 0 On differenttiating 5 with rreiajpenzt a and EquatViVmg the result In N we gait 3 p5 V ay 50 Mm 5JMm yields g y Hi I M ay J MdJdyi 6 prmrided the intagrarnd here is 1 fungrti m m ty nf will has true if the derivat ivE If this inmgjgriandi with TEEpEC17 than 3 i5 1 and since lfh Aderhrative p u6St5i131 is 5 s Aa g 51 A N Mdx Jwdx 69 By 3 61 ay an 31 Mix H Sy x w w P P quot 339 an appeeall trim 01111quot assumptim 4 cm4mfpllete5 tha argumemL I The readar ShrmI d ha wan that rqL1aEi wn 3 is hru w h EnEver Lb r1 th sides ami5t anti arr mnIinu M5 and aha tI1me c ndimimn5 are smiJ5 rJ by sa mI15E all funati n5 that an l i1u Ir m 3rTi5n irl pr E i tr Cmr b lacnilae t hypDthBsi5 Ehmugh m thi chapter SEE II39mc rs pa ir 3graph in 5E9L TiID39 T is that 3 me fun ti ns W discL1ss azrz auf cieiltiy cnn39Iinu1J5 and diiffaar1n ti1mrIw2 in guarw139IE EhE validity tzuxf thm 3pEralium15 we p rfmrmz Lin 1hEm FIRST ORDER EQUATIONS 53 In summary we have proved the following statement equation 1 is exact if and only if BM8y 8N8x and in this case its general solution is f x y c where f is given by 5 and 6 Two points deserve emphasis it is the equation f x y c and not merely the function f which is the general solution of 1 and it is the method embodied in 5 and 6 not the formulas themselves which should be learned Example 1 Test the equation e dx xey 2y dy 0 for exactness and solve it if it is exact Here we have M e and N xe 2y so 8M y 8N d y ay e an ax e Thus condition 4 is satis ed and the equation is exact This tells us that there exists a function f x y such that 3f lt9f d y ax e an ay xe 2y Integrating the first of these equations with respect to x gives f feydx go xey go S0 a 5 xe s y Since this partial derivative must also equal xe 2y we have g39 y 2y so gy y2 and f xe yz All that remains is to note that xe y2c is the desired solution of the given differential equation PROBLEMS Determine which of the following equations are exact and solve the ones that 3I C 3939 E 39 39 139 2 xdyydx0 sinxtany 1cix cosxseczydy 0 y x3dx x y3dy O 222 4x 5dx 4 2y 4xydyp y ycosxydx x xcosxy dy 0 cosxcoszydx Zsinxsiny cosydy 0 sinxsiny xe dy e cosx cosy dx miA1 ERENrtAL EJuamDrs E i51J1n d sinidy E 9 1 yjir l may p pm HI 2x 39 uznznm i3tt14yZ Sin 1quot dy U y 1 e siny 5 L4ty3 J Es U395 CL 13 y adx P 5 0 6 0 av 14 31 y 0 tlfx I113g y Aryjdx J lwzzrgx vxy dy D 16 g as y cscgxx lrye cscy cm 3 cmx U B 1 yg H 23 cnszxdy 2 U 3 J M M P PO 2 39 quot p P l gyjdx 0 F Pi 2 0 SDEVE ex My 4 as an EIILHCI raquatEion in two W jf an l recnn ci1c the irEsuJVt5i 21 Snl1rE 4 3 X mjy ix jy T if p 4x 93 dye g J D E as an E zlE39t Bql1atinn b as a humungene us equat39inn Firuzli 1he mine of H for which each GTE the quotf mng rEquaiI39rinns is 33131 and Salve thg eq uatimnA far that ma Iue uf 32 3 fxyl x3 r D M IE2 yam 08 HKEE 39 The Vreadser P p rr1babl r n miced that exact di ElquotquotE1 1Iia1 equatimfls are mmpaIativcjy Zfi I39E for exactnEss depEnds DH 51 prmige bal3m1ce in the farm f the equatiun and is vsasihr de5truyed by minm changie5 in this form Under th l cimumstances T is rea50n3b4le m ask whethm e3auVm e qua1imns art wrzsrth discus5ing at ML Irm the p resen39t 5 ecVtiran we 5l1al1 try ta mnvitnca the i39EHdEl1quot that they a1394Ei FIRST ORDER EQUATIONS 55 The equation L ydxx2y xdyO 1 is easily seen to be nonexact for 8M8y 1 and 6Nax 2xy 1 However if we multiply through by the factor 1x2 the equation becomes y 1 39 which is exact To what extent can other nonexact equations be made exact in this way In other words if Mxy dx 1 xly dy 0 2 is not exact under what conditions can a function ux y be found with the property that uMdxNdy 0 is exact Any function u that acts in this way is called an integrating factor for 2 Thus 1x2 is an integrating factor for 1 We shall prove that 2 always has an integrating factor if it has a general solution Assume then that 2 has a general solution f x y C and eliminate c by differentiating Sf Sf dx d O 3 ax ay y It follows from 2 and 3 that y c9f8x dx N afay so 8f8x 8f8y 4 M N If we denote the common ratio in 4 by ux y then 8f 8f M d N ax u an By a On multiplying 2 by u it becomes uM dx MN dy 0 8 E9 1 dx f d 0 8x 8y y 0139 56 DiF FEREHT39i39AL EQuATI Hs whilch is EKHE1JFC This argumant slwws tlmat if has a germerzal 5Dlutimni than has at least mm integrating farmI W Acrtually it Has irn nitely mVany im tampgrzaIing fkacmrsz frzar Ff is any fumctinn Gff than p t aw Fm dr 2 a UFf err 50 pLF is also an inI egrati11g fact far 2 Our di5cussi0n sea far has mIt mnsidered the jpracticxai p1quot bEm mvf nding integrAating 1factmrsi In g Imra this is quite di tzzult Thuare Aam a fEw cases hDwcmer in which fmrmali prnceduras are avai lable4 Tn see haw these prmed4ures ax1352 we C0nsi d r the cnncdTiti4nn that gs he an intcgrating factmr m 2 Sim E aWLW ay EL E If we write this mum Awe z1bteain IL39lI39 1 61 4 SLM N NiM i 5 E ax Va 5 C It ajp ars tfhait WE have 139amp vJuced39 th pmblem f S hFi g the Dj inary di Eri39Intial equatinn C 2 to much mm di nul l prmbiem nf 5arlving the partiam di39 erAampntia equat39inn On the other hand we have m used fur the getneral 5 zaluIirn Inf 5 since any p 1 IE39il3LIIl Iquot E1utinn wiM serve mm puirpcsse And from Apnint 01 viasw 5 5 mare fmi uil than it lcn kcs Supsc for instame that 2 has an intv2grati4ng factor pm which is a ifunmimfI Inf 1 alone Then 5 ufSx duf and 5 L fn35v 6 30 5 can be w1ritI 3n in EhE farm Idgu 3M8y Bw ax 39 P 7 P 3 5 u p 4 N S ince the lexft 5 ME nf P is a funcztiim inly Hf 32 the right is ralsrm If we put N 39 than 6 bacwinme3 flj an 3 pr it FIRST onoen EQUATIONS 57 or d10g M I dx gx so logu gx dx and u ei3xdX h This reasoning is obviously reversible if the expression on the right side of 6 is a function only of x say gx then 7 yields a function 4 that depends only on x and satis es equation 5 and is therefore an integrating factor for 2 Example 1 In the case of equation 1 we have 8M8y 8Nax 1g 2xy 1 2xy 1 2 N x2y x xxy 1 x D which is a function only of x Accordingly ej 2xdx Zlogx 2 1 3 x is an integrating factor for 1 as we have already seen Similar reasoning gives the following related procedure which is applicable whenever 2 has an integrating factor depending only on y if the expression aMay aNax M 8 is a function of y alone say h y then H 61 hydy 9 is also a function only of y which satis es equation 5 and is consequently an integrating factor for 2 There is another useful technique for converting simple nonexact equations into exact ones To illustrate it we again consider equation 1 rearranged as follows xzy dy x dy ydx 0 10 The quantity in parentheses should remind the reader of the differential formula dy xdy ydx 11 x x2 53 m 2 AL EQUATIDN5 wV1 mich suggEs ts dividinig 10 thmvugh x x IranS39fDrms the equaVti mn into y dy d yf x 0 an its g11EnErali sa0lutio4n is evi en 3 1 3 By Iill In amp et we hzawme fx u d an integrating facmr far 1 by ncntiwing 0 it the mmAbAimatirm I afy y zu and using M in BI39p1i39i this b3ervatium 9h fnlInwin are mm mher diemntial fQrmquotu1as that are amen useful in similar circumstanee5 q 1 it 9o p3 p gj d5 Pm my 11quot J dxy dy J q 13 d 9 3391 3I Py 4 e di v 14 JLTV n l E Tf P l dtan l g l P k y I y E h E I J l aw We see fmm t hese fr39muEas thsat the uer3r siimple diJerenIIial El39I 1IigtE II 1 dong 16 ydgr E has 1fx 3 A1fyE A 1fr3 f and lfxy as integrming factnrs gmd thus can be s mh ed in this manncr in 3 Variety nzmf wa 5 EIEIEPIE 2 Fimzl the shape Elf H curved mir mr Efu h that 1igh I frmn H SDUIEEE at the s39nrfigin will he re ectad in a bearm E31f ra5 parmiel to the xaxi5 By 5ymm trr th mir1 m will hIaHgte the L h ip Bf M143 S ff Uf mv uEu1inm ganerarted gr re4v lving a cu rv5 APB Fig 0 U about the raami5 FIG 13 FIRST onor52 EQUATIONS 59 It follows from the law of reflection that O 6 By the geometry of the situation lt35 3 and 6 or lt12 23 Since tan 9 yx and Ztan3 t 9 2 an an 1 tan2B we have 2d dx 1 dydx Solving this quadratic equation for dydx gives xlVx2y2 dx y xdx ydy lVx2 y2dx X x 0139 By using 14 we get dx2 yz 2Vx2 y2 iVx2y2x c On simpli cation this yields dx S0 2 Zcx C2 which is the equation of the family of all parabolas with focus at the origin and axis the xaxis It is often shown in elementary calculus that all parabolas have this socalled focal property The conclusion of this example is the converse parabolas are the only curves with this property PROBLEMS 1 Show that if lt9M8y 8NaxNy Mx is a function gz of the product 2 xy then 1 eIgzdz is an integrating factor for equation 2 2 Sove each of the following equations by nding an integrating factor a 3162 39 y2dy 39 Zxydx 0 b xy 1dx x2 xy dy 2 O c xdy ydx 3x3yquotdy 0 d exdx e coty 2y cscy dy 0 e x 2 sinydx23xdcosy0dy 0 f ydxxZxy y g x 3y2 dx Zxy dy 0 h ydx 2x ye dy 0 i y logy 2xydx x ydy 0 j y2xy 1dx x2 xy 1dy O k x3 xy3dx 3y dy O zN 1a39iFFEnEVruquotrnaL EQUATIONS Under wl39mt iVrcumstancrE5 will equatirn 2 Thaw an integra1ing fa t r that is 3 funst4iaan of the sum 3 I TEA S lfve 1113 fullwring Eq1iati m 1 by using the di arenEi al ftrmulas 12 I6j Ia 0 1 lm bl yd my xvquot d39n o my 153 yquot pTC air 61 U rider 9 39r E 139 dy J 2 g it U3 ej r my Jq 6 5 Mi Mix E212 Slit am my m ap Lt 39 4 u JV 12322 dy U U 1 iv y Pv ir ti y it an my 3s l W dy U U 23 EliI 39 Idle 0 6 dy j 5 rq 5 Su Ive H1 f 39H 39Wi1g equati n by Amaking 0 k 5ub3titmtim1 2 r r 39 nr l xzquot and Eh sing 3 C nvenient 39equoti l1ET for 911 iJmmw til 33 A 1 Iectargular cmVrdinate5 Hmr 10 Q 1 D Find the curv AP in Emmpe 2 by using psolar rDrdiVnAa ces instead Hf 1 LINEAR The nmst iLmpmtan1 tye emf diL crm39tiaI tqVuati1un iLs m Einear equarmrn in which the d1Vriva tiv e Px1 highest rdrr is El linear f u1391mi m of the lrawer 0rder dEriAvkaVtives Thus genErai rst 3939TdET li EElquot39 eqL1ati 39rn is 3 Fwy am the general 5 end order linear equatisnn is day dxi and ms D11 M1 E 1merstmmd that 1111 uef7 vi ents cm t he in tl1EEE eacprE35imms namalylt pfxfh qz rI etc are fu4nc1 Aimns pQ cane Elms pmsent mrmern is with Ih igsneral first Drder lincar 6ql1LE fiU lh which we write in the standard farm I 1 mn qu mw PwnQmy my 1 FIRST ORDER EQUATIONS 61 The simplest method of solving this depends on the observation that d dy dy 21 xe quot y e quotd yPe quotquot e quotquot quota x Py 2 Accordingly if 1 is multiplied through by ei P d it becomes I d xeI39Pdxy QefPdx Integration now yields e quotquot y IQeIP dx c so p yequotIP fQe quot dxc 4 is the general solution of 1 d 1 Example 1 Solve T E y 3x This equation is obviously linear with P 1x so we have 1 J39PdxfJCdxlogx and e5Pquot e quotx On multiplying through by x and remembering 3 we obtain d 32 dxry x S0 xyx3c or yx2cxquot As the method of this example indicates one should not try to learn the complicated formula 4 and apply it mechanically in solving linear equations Instead it is much better to remember and use the procedure by which 4 was derived multiply by ef P quot and integrate One drawback to the above discussion is that everything hinges on noticing the fact stated in 2 In other words the integrating factor ei Pd seems to have been plucked mysteriously out of thin air In Problem 1 below we ask the reader to discover it for himself by the methods of Section 9 PROBLEMS 1 Write equation 1 in the form M dx N dy 0 and use the ideas of Section 9 to show that this equation has an integrating factor it that is a function of x alone Find it and obtain 4 by solving uM dx uN dy 0 as an exact equa on FIRST ORDER EQUATIONS 63 a Find the amount of salt in the tank when the brine in it has been reduced to 20 gallons b When is the amount of salt in the tank largest 11 a Suppose that a given radioactive element A decomposes into a second radioactive element B and that B in turn decomposes into a third element C If the amount of A present initially is x if the amounts of A and B present at a later time t are x and y respectively and if k and k2 are the rate constants of these two reactions nd y as a function of t b Radon with a halflife of 38 days is an intensely radioactive gas that is produced as the immediate product of the decay of radium with a halflife of 1600 years The atmosphere contains traces of radon near the ground as a result of seepage from soil and rocks all of which contain minute quantities of radium There is concern in some parts of the American West about possibly dangerous accumulations of radon in the enclosed basements of houses whose concrete foundations and underly ing ground contain appreciably greater quantities of radium than normal because of nearby uranium mining If the rate constants fractional losses per unit time in years for the decay of radium and radon are k 000043 and k2 66 use the result of part a to determine how long after the completion of a basement the amount of radon will be at a maximum 11 REDUCTION OF ORDER As we have seen the general second order differential equation has the form F xyy y 0 In this section we consider two special types of second order equations that can be solved by first order methods Dependent variable missing If y is not explicitly present our equation can be written fxry39y 0 1 In this case we introduce a new dependent variable p by putting d y p and yquot 1 2 This substitution transforms 1 into the rst order equation dp fxp1 0 3 x If we can nd a solution for 3 we can replace p in this solution by dydx and attempt to solve the result This procedure reduces the problem of solving the second order equation 1 to that of solving two rst order equations in succession DiFFE7HErfrML iElt3iimTI rIEH5 I Iii Emimple r Salve xy y 0 31 The variab1re 3239 is miaiiinig fmm this equatimi 54 72 iraciinvcesi it to E p 311 or dp l an 4 3 4 T JP J which i H E L Q11 sawing this by the mcthnd of Eamon 10 we mbttain SD 1 1 y 2 Jar quot rmcz E2 2 is the dieaireid 5 0lu1iDr1 lndepemijient i 1j hIE imissing If 1 not EKpli39iiily pirega nt our s c ndx wider e quat imn can be written jmquot yquoti P 0 4 Here we mtiriciiuce DU139 msw dEpE11diEnt 39wquot39aaI39iah E p in the samie way 1 but this time wc exiprgs5 y in termsi f ai deriivativie with ricsipsct 120 L y Q 0 Tlriis engaibfies us In write 4 in the form gvp P dy and fmim this p irit on we pirinceicd as above E lvi g two rst nrdezr eiquaticsns in suic essiii y L 39 p and Empie P Solve y kgy With the raid of 5 we can write this p the fnrim k yi 0 U1 p iEjy39 dquot U In teg1quotalim ni jii id l E EFF FIRST ORDER EQUATIONS 65 A second integration gives sin i lkx b so yasinlkxb or y Asinkx B This general solution can also be written as J y c1sinkx C2 cos kx 7 by expanding sin kx B and changing the form of the constants The equation solved in Example 2 occurs quite often in applications see Section 5 It is linear and its solution 7 will be tted into the general theory of second order linear equations in the next chapter PROBLEMS 1 Solve the following equations a yy y 2 0 6 Zyyquot 1 y b xy y y 3 f yy y 2 0 C yquot kzy 0 g xyquot y 4x d xzy 2w y 2 2 Find the speci ed particular solution of each of the following equations a x2 2y y Zxy O y 1 and y 0 when x O b yyquot yzy y 2 y and y lwhenx 0 c yquot y39e y 0 andy 2 whenx 0 3 Solve each of the following equations by both methods of this section and reconcile the results a y 1 y 2 b y y 2 1 4 In Problem 58 we considered a hole drilled through the earth from pole to pole and a rock dropped into the hole This rock will fall through the hole pause at the other end and return to its starting point How long will this complete round trip take 5 Consider a wire bent into the shape of the cycloid whose parametric equations are x a6 sin 6 and y a1 cos 9 and invert it as in Fig 10 If a bead is released on the wire and slides without friction and under the in uence of gravity alone show that its velocity v satis es the equation 4av2 gs 32 where so and 5 are the arc lengths from the bead s lowest point to the bead s initial position and its position at any later time respectively By differentiation obtain the equation and from this nd s as a function of t and determine the period of the motion Note that these results establish once again the tautochrone property of the cycloid discussed in Problem 65 1 ED SJquothTI H A Px 0 Px We mm discuss slagm ral atp pltitatim1s leading tn di etrernttitatl e qtua39tin ns IhaI can be S w d by tm t mmhndts of this cjhaptert Emmple Find the shapa a5 5umeuE by a il ibi chain SuEpEaI39dEd betw ein Ewe points 31Id thantging IJJIi dBI its Dw39I I2 weitghitt VLet the y axis pass tthmugh the 1uwe539t pain at the chain X lat Net 3 the arc1a1ngtiht fmrnt this p int In a varialzalte paint ry i and last 39w be the tlinaar densitgr cf the C39JhHi We Gbtaint the aquarium aft the wzurve fimm thct fan t that the pmtimrm anf tthe 1hai1n betwean the I w stt pnitrtt and t39xyjI is in equtili3Jrium Un t the a lin n of three f r1rces the hnIriimntai tcnsi n p at tthat l awe5t quotpainttt the vgartiabtlle ttentsiun at I whin acts aiming the ttangtent tzaet ausa of the t E3ti39i1ity mt tttc vchai and a dtuwnwardt farce equal tn the wcight t lff the chain between tttmse WYit p1in39t5t i quating that h t iE tal cnm potnent of P tn Tu and the verti ctaI Eamtp metnt nf T to the weitght Bf the Chain giv5 5 Tees 6 P and Ttsin H J ws 9 It fnligw nwmt the ftrsztt of tthese requatim1t5 tthat Tail 9 T3tan H 311 FIGIJ H FIRST ORDER EQUATIONS 67 SO 73y f wsds 0 We eliminate the integral here by differentiating with respect to x 78 iJwsds 1iJwsdsg WSV1 4 quot2 TBYquot WSV1 39z 1 is the differential equation of the desired curve and the curve itself is found by solving this equation To proceed further we must have de nite information about the function ws We shall solve 1 for the case in which ws is a constant wo so that yquot ax1 yr a 11 lt2 7 Thus On substituting y p and yquot dpdx as in Section 11 equation 2 reduces to dp a dx 3 WW We now integrate 3 and use the fact that p 0 when x O to obtain log p 1 477 2 ax Solving for p yields dy lax 4 dx 2e e If we place the xaxis at the proper height so that y 1a when x 0 we get y 1 e e quotquot lcoshax 2a a as the equation of the curve assumed by a uniform exible chain hanging under its own weight This curve is called a catenary from the Latin word for chain catena Catenaries also arise in other interesting problems For instance it will be shown in Chapter 12 that if an are joining two given points and lying above the xaxis is revolved about this axis then the area of the resulting surface of revolution is smallest when the arc is part of a catenary Example 2 A point P is dragged along the xyplane by a string PT of length a If T starts at the origin and moves along the positive yaxis and if P starts at a0 what is the path of P This curve is called a tractrix from the Latin tractum meaning drag 1ttFFEtREHTIsL E quotLFFtTUN5 Fl pW It is easy to sac fmm 15 that thus di emntis sqtttstmn f ttm path p E p E H sspatating vsl iames a1rtIi inttzgrattingt and using thtr fact that y II when T A 1 AWE trtd that is the squstinn at the tft7Et I3tI iI This curve is art crrnssidsrsts s impnrtanre irt E m ift bssst1se the ttrummstsatshspeds ssu rf3EE DhtaiL391Ed by rsml39ving sfswt thus yasttsis 0 is T lme jt far Lmthass hssvstty ts version of nnn4Euslides gE imrstrmt since tthe zsum at the snlglss if any tritsnggtes dquotrsssn rm ths s t1rzfsse is lsss tthsm Altsn in the ttsrntsst nf di t139IEntiaI gensmstry this sutsfscc is tst1sti E P EH f EptEIEFE39 bsssustci it has constant negsttisrs rl urttiitltr as Dpfptsssd tn ths constant pns Etiw EUtW393EtJ t39E of s swhtere Example Z t rsb l si t starts at ttlte urigin and rtms up the yssztist wi tht speeds 1 At the same time 1 dog running switht speed p starts at the p ittt scUI snd purssu sss the ra rit What D the at the atflogg At tims rt measurEd fmm the instant hth stsrtt the rsbibitt M1 has at tits pnittt tA sttjI and th dug at ry F39igtt IE1 Sir1r E Iihe Hm DR FIRST ORDER EQUATIONS 69 M KR 0at NV 50 FIGURE 16 is tangent to the path we have d at 3 y x or xy y at 4 To eliminate t we begin by differentiating 4 with respect to x which gives xy a 5 Since dsdt 2 b we have dt dt ds 1 1 I 2 dx ds dx b y 6 where the minus sign appears because 3 increases as x decreases When 5 and 6 are combined we obtain the differential equation of the path xw W1 o39gt2 k 3 lt7 The substitution y p and y dpdx reduces 7 to d d V1p2 X and on integrating and using the initial condition p 0 when x c we 7 1sIFEm3rrIL E 39LJ ATI HS n uI thEI39I H in in p Vv V 1T 4 pi Rug Thi can rea i39Iiy be silnad Emquot pg yie1ding s W En P E air It 2 t Cs Xx i In mr der in csn39t4inue and ml y A35 3 39f39IIl11E39tiv l3i39I1 of 1 WE must Ahgws fur 1er infurrm ticm ab ut We ask the I39E dEfl39 In 3tp1me mm f the p ssi7bilitiEs in TFquotr hiEm 8 ExampI1e The yeaui5 andf the ine 1 at are the banks of a river whuse curzmnt hag unfmquotm Speed 2 in the negative y dEimcIicnn boat Enters th I17wI39 at the puinii arid hgaills KdireE 3r ts3ward 39t39h 139igi witgh sp ed 3 relative 11 the water What is the palm of the bum The Iu39rI139p InEn t5 Of the b at 5 vellnnsiw 1 are z 2 TTb 105 9 and Ad a sin 8 as E SE dr ib ms 9 Vb4wrV I1 1 dy as bsin o an bJLy is 32 a 1 by till z 3quot it Lii 8 1 71 FIRST ORDER EQUATIONS 5 This equation is homogeneous and its solution as found by the method of Section 7 is Cky 1x2 y2 xkl where k ab It is clear that the fate of the boat depends on the relation between a and b In Problem 9 we ask the reader to discover under what circumstances the boat will be able to land and where PROBLEMS 1 2 In Example 1 show that the tension T at an arbitrary point x y on the chain is given by wo y If the chain in Example 1 supports a load of horizontal density Lx what differential equation should be used in place of 1 What is the shape of a cable of negligible density so that ws O that supports a bridge of constant horizontal density given by Lx Ln If the length of any small portion of an elastic cable of uniform density is proportional to the tension in it show that it assumes the shape of a parabola when hanging under its own weight A curtain is made by hanging thin rods from a cord of negligible density If the rods are close together and equally spaced horizontally and if the bottom of the curtain is trimmed to be horizontal what is the shape of the cord What curve lying above the xaxis has the property that the length of the are joining any two points on it is proportional to the area under that are Show that the tractrix in Example 2 is orthogonal to the lower half of each circle with radius a and center on the positive yaxis a In Example 3 assume that a lt b so that k lt 1 and nd y as a function of x How far does the rabbit run before the dog catches him b Assume that a b and find y as a function of x How close does the dog come to the rabbit In Example 4 solve the equation of the path for y and determine conditions on a and b that will allow the boat to reach the opposite bank Where will it land 13 SIMPLE ELECTRIC CIRCUITS In the present section we consider the linear differential equations that govern the ow of electricity in the simple circuit shown in Fig 18 This circuit consists of four elements whose action can be understood quite easily without any special knowledge of electricity A A source of electromotive force emf E perhaps a battery or generator which drives electric charge and produces a current 1 Depending on the nature of the source E may be a constant or a function of time T2 DilPFERElfT39Ih E ErCrUA1 IDH5 5 E I FIG IE 0 rLe3istr f rampsisVtanVre R which pp ses the C1LIll39E39EquotEI39t by pmdumgzimg a dmp Emf Elf magnitudse 0 eq39uatian is caMeVJ Q hm5 IawQ2 An inr1lur mr Uf imductanc L which UeppUse5 any change in the cuA4rmnt by pmdu cir1g Ma dmrp in d P magnitudVe Y L L as D cagpacit lnr mt u0ndE4ns er caf capacitanc which stares the charge The harg accumquotula ted by the capaacitcjr 1rltesists tfh in ow Hf dIfifiti l azzfnargci and the dmp in Egamf arising in this way is EU0 rg Simyfrm Uhm r391TT 1354 Wm a G rmam ph 5i is39t whm 2nlgpr 5igni c ant acrmt39Imra inn In SE39iE EE was his di5 uwur1quotjr of 1l1e law srtamd ahuzwe When hue ann unced it in 181239 it 5 cmedi ms gmrcl in he tmu and W35 mall hE39HEquoth39E39l J Oham wag av39unsid39red unw ablte 3ms1uie at this and was sun badly 39nmated rIham he rasiig J1e his p rUfE55m1quot5hip art Cu mgne and ived tm st mtrai yaars rm 39E39ZiyV35 i Mi1 1iiIjF ami p391IIrquotIn39EI quot39 befmre it wag Lnmugnienl s he was right Guru of pupils in Culngruec was P39 1er D39irfichim when lam bcramET one M the mmt eminem G srrman mahcma Lirim5 iE th ninrEcnth emur3u FIRST ORDER EQUATIONS 73 T Furthermore since the current is the rate of ow of charge and hence the rate at which charge builds up on the capacitor we have 1 9 dt Students who are unfamiliar with electric circuits may nd it helpful to think of the current I as analogous to the rate of flow of water in a pipe The electromotive force E plays the role of a pump producing pressure voltage that causes the water to flow The resistance R is analogous to friction in the pipe which opposes the ow by producing a drop in the pressure The inductance L is a kind of inertia that opposes any change in the ow by producing a drop in pressure if the ow is increasing and an increase in pressure if the ow is decreasing The best way to think of the capacitor is to visualize a cylindrical storage tank that the water enters through a hole in the bottom the deeper the water is in the tank Q the harder it is to pump more water in and the larger the base of the tank is C for a given quantity of stored water the shallower the water is in the tank and the easier it is to pump more water in These circuit elements act together in accordance with Kirchh0 s law which states that the algebraic sum of the electromotive forces around a closed circuit is zero3 This principle yields E ER quotELquotEC0 or d1 1 ER1 L 0 dt CQ which we rewrite in the form d1 1 L RI E 1 dt CQ Depending on the circumstances we may wish to regard either I or Q as the dependent variable In the first case we eliminate Q by differentiating 1 with respect to t and replacing dQdt by 1 d2 d1 1 dE L R 1 2 dt2 dtC dt 3Gustav Robert Kirchhoff 18241887 was another German scientist whose work on electric circuits is familiar to every student of elementary physics He also established the principles of spectrum analysis and paved the way for the applications of spectroscopy in determining the chemical constitution of the stars 74 iDi 7EREHTEALECIHATID15lE In the second ae we simply replare E by dz 06 5 R 0 LRampr EQEw 3 WE shall mnsid er thase secnynd nrd Er linear equaIi0 n5 rm mme detail laIer Dutr E lquotlCE139quotI1 in this 5en39ti n is primmriIy Vtmith the rst nrder Ivi EKm f equa nn M at P btEi Ed fmrn 1 wxhen nu v2apauitsr is presEnt Example 1 Salve EUa39im1n 4 fiat the E356 in wI1iI39h an initia 1 Iiillrmn z IQ is nwing and as mnstant emf En is impressed 11 the 1a1rcuit at time I 2 D For I 3 D nur Eqllati fl is H L d M The variaI1E5 an be separated iE Wding quotTL tit On VinIegraIing and using the riinitial Emuditimm I In whrn I D we get ing ha RI I l g g E RIB EU I s fm Eli Not that the urrent I 3 IquotlSi55 Hf a Sfe tii39E5 I part Em 0z and a rrn539 39Enr part L3 EwJ R e quotquot tha39t appramaihe5 zmwn as I tinuraas3s CQnsEqu en 39VV Ohmquot1 aw Ea b nearly truc fair large L WE amt b5ervc that if 15 s P than P E J rL39 i I 1 E and if 59 0 thEn I 1 In Extamp1e 1 with if 11 and El Shaw that the rzurrmt in the circgtuit lhuilila Vup Em half its 39th m39EfiEa main1um in L Egg 2fR SEEU d 75 FIRST ORDER EQUATIONS Solve equation 4 for the case in which the circuit has an initial current In and the emf impressed at time t 0 is given by a E Eequotquot b E E sin wt Consider a circuit described by equation 4 and show that a Ohm s law is satis ed whenever the current is at a maximum or minimum b The emf is increasing when the current is at a minimum and decreasing when it is at a maximum If L 0 in equation 3 and if Q 0 when t O nd the charge buildup Q Qt on the capacitor in each of the following cases a E is a constant E0 b E E05 3 E E cos wt Use equation 1 with R 0 and E O to nd Q Qt and I It for the discharge of a capacitor through an inductor of inductance L with initial conditions Q Q0 and I 0 when t 0 MISCELLANEOUS PROBLEMS FOR CHAPTER 2 Among the following 50 differential equations are representatives of all the types discussed in this chapter in random order Many are solvable by several methods They are presented for the use of students who wish to practice identifying the method or methods applicable to a given equation without having the hint provided by the title of the section in which the equation occurs 1 hi a5 1 RGEc392a E39S3 9 quot Equot S yy y39 1 xyy y 2x3y1dx2y 3x5dyO xy39 Vxz yz yzcix x3 xy dy x2y3 y dr x3y2 x dy yy y 2 Zyy 0 xdy ydx xcosxdx xydy xzdy y2dx equot 3x2y2y ye 2xy3 y 2xy3920 x2ydxxdy xy y xzcosx 6x 4y 3dx 3x 2y 2dy 0 cosx ycix xsinx ydx xsinx ydy xzyquot xy 1 y2equot cosxdx equot xyequot dy 0 y logx y1logx y 12 y Zxy e yz 3xy 2x2 dx x2 xy dy T63 DiIIquot39FE HEHT1AL Emm1mHs p Wu I2r B 2 439 22 E sin E rI7r5 yquot d y sirl xy dx J Eirury v 1 r2yquot xyquot U v x y idy 2 E v A Equotquotl xequot yamp 39dyV r E v 1quot A xjfdy r Jrquot A s p p tan 1 F 13 X p s s 2xyEaTvEI3 0 s PYs E 3 T jivrri 1 2 M m 39 y E quotquot39 H E 1quot 3 31 Iggy cit riy Will 3 05n l o pn it n dn sin Rn n m vn E 21 39 j I H3 I W p m xjs1 yi1f xiv U 35 j y I3339 2324 k g y y3 e 39 x tquotd U bH cr2 lyquot j Lrli 1quot jm32y 1 dx 2x 3y 1 i EW1 ax Sb dy 2 LL 3x it r3Equot 5invdrt y M3 AV u p 3y39 yEdz II Bxy 32 dy U 43 Jcly E xi Qty 6C xyquot Rb VyllDgx 45 C s a V P c an 1 3 If 5mny 1wgJr Fa y a J2yquot h U 4 quot A3 J 1 11 1 d K 3 y s 49 P 3yy39T msbz 50 S 1 E01 mushy FIRST ORDER EQUATIONS 77 51 A tank contains 50 gallons of brine in which 25 pounds of salt are dissolved Beginning at time t 0 water runs into this tank at the rate of 2 gallonsminute and the mixture flows out at the same rate through a second tank initially containing 50 gallons of pure water When will the second tank contain the greatest amount of salt 52 A natural extension of the rst order linear equation y px qxy is the Riccati equation y px qxy rxy2 In general this equation cannot be solved by elementary methods However if a particular solution yx is known then the general solution has the form 39x y1x zx where zx is the general solution of the Bernoulli equation 2 z q 2ryz rz Prove this and nd the general solution of the equation yr x3y2 x5 which has yx x as an obvious particular solution 53 The propagation of a single act in a large population for example buying a Japanese or Germanmade car often depends partly on external cir cumstances price quality and frequencyofrepair records and partly on a human tendency to imitate other people who have already performed the same act In this case the rate of increase of the proportion yt of people who have performed the act can be expressed by the formula E lt1 ygtsltrgt 1y gt 4 Count Jacopo Francesco Riccati 16761754 was an Italian savant who wrote on mathematics physics and philosophy He was chie y responsible for introducing the ideas of Newton to Italy At one point he was offered the presidency of the St Petersburg Academy of Sciences but understandably he preferred the leisure and comfort of his aristocratic life in Italy to administrative responsibilities in Russia Though widely known in scienti c circles of this time he now survives only through the differential equation bearing his name Even this was an accident of quothistory for Riccati merely discussed special cases of this equation without offering any solutions and most of these special cases were successfully treated by various members of the Bernoulli family The details of this complex story can be found in G N Watson A Treatise on the Theory of Bessel Functions 2d ed pp 13 Cambridge University Press London 1944 The special Riccati equation 2 by2 arm is known to be solvable in nite terms if and only if the exponent m is 2 or of the form 4cZk 1 for some integer k see Problem 478 FIRST ORDER EQUATIONS 79 which simpli es to we note that dmdt is positive or negative according as the body is gaining or losing mass and that w is positive or negative depending on the motion of the mass gained or lost relative to m The following problems provide several illustrations of these ideas 55 A rocket of structural mass ml contains fuel of initial mass quot72 It is red straight up from the surface of the earth by burning fuel at a constant rate a so that dmdt a where m is the variable total mass of the rocket and expelling the exhaust products backward at a constant velocity b relative to the rocket Neglecting all external forces except a gravitational force mg where g is assumed constant nd the velocity and height attained at the moment when the fuel is exhausted the burnout velocity and burnout heightquot 56 A spherical raindrop starting from rest falls under the in uence of gravity If it gathers in water vapor assumed at rest at a rate proportional to its surface and if its initial radius is 0 show that it falls with constant acceleration g4 57 If the initial radius of the raindrop in Problem 56 is r0 and r is its radius at time t show that its acceleration at time t is 5 lt1 7 4 rd 39 Thus the acceleration is constant with value g4 if and only if the raindrop has zero initial radius 58 A spherical raindrop starting from rest falls through a uniform mist If it gathers in water droplets in its path assumed at rest as it moves and if its initial radius is 0 show that it falls with constant acceleration g7 59 Einstein s special theory of relativity asserts that the mass m of a particle moving with velocity v is given by the formula m on V1 9 U2C2 where c is the velocity of light and m is the rest mass 6The experience of engineering experts strongly suggests that no foreseeable combination of fuel and rocket design will enable a rocket starting from rest to acquire a burnout velocity as large as the escape velocity This means that singlestage rockets of this kind cannot be used for journeys into space from the surface of the earth and all such journeys will continue to require the multistage rockets familiar to us from recent decades D IFFEREN39TMmL EunTmuN5 3 p they partir1e starLs fnrm uresat Empty 5pace and muves for a lung time unider the in uence caf 3 un5tant gravitaeIinnaI eId i nd L as a functinn orff by ta39kLingA W U and Show thai nu quot3 c as E a man he VI et V m mu be the increzass in tha mass of the partir1e If the umrrgte5pmnd39mg incmasse in its EmErgy is tak n In be I211 wurk dnn m it b1 the pre1railiVng farce F 56 that E I Fir I imujwx J udm u D 4 395 Tm wmI39ijr thquotat E f p iIr E Dediu4ce Am fmm am 1 Enrim Fermi has 5ugg wed that th Ph Elquot1 1Eifl n decribcd hmquotE irarn5ferr39eI 7 In rim E3513 1 zcharg ed par39lils f in39ter teEHa ase1grate d by the magntEu glcis Elf 5etmrs1 can anrgnzmlnt in part far the migin mi pirilmargr r tsmic rag395 CHAPTER 3 SECOND ORDER LINEAR EQUATIONS 14 INTRODUCTION In the preceding chapters we studied a few restricted types of differential equations that can be solved in terms of familiar elementary functions The methods we developed require considerable skill in the techniques of integration and their many interesting applications have a tasty avor of practicality Unfortunately however it must be admitted that this part of the subject tends to be a miscellaneous bag of tricks and conveys little insight into the general nature of differential equations and their solutions In the present chapter we discuss an important class of equations with a rich and farreaching theory We shall see that this theory can be given a coherent and satisfying structure based on a few simple principles The general second order linear differential equation is dzy dy 5 Pxd x Qxy Rx or more simply y Pxy Qxy Rx 1 81 32 eirFEseHTtst EeeaTrens As the netatien indiealtes it is luncle39rstteecl thati l r Qlx and Rx are hlnteltlens ei at alene er perhaps eenstants It is clear that lie less ef generality results frem talting the eeet lleient erf y te he 1 since this can allways be aeeempClisl1ed iy lliirisient Eqtiaftiens elf this kgincll are ef great signi elanee in physicist eslpeeiatll3 39 in eenneetien with vi39hratiens in rneeharlies anal the them erl eleetrie eireuitsi In aclIlitierI as we shall see in later ehaptlers rnany pl3939lfUllI39lii atnd beautiful ideas in pure niathe rnaties shave grenrn eat at the hatred elf these E39q39Ll391 tiUI39lS We should net he I39l39lil5lEl l the fast that first eraser linear etiltlfttiEllie are easily sellved lay means eli ferrnutlas In general P eannet he lsietlvedt explieitly in terrns elf ltnewtn ellernentar funtetiteins are even in temis of it lliCEt39ll d ir1tegratiens Te nd serlt1tiet1s it is eernmenly neieessary tie resert te inr nilte preeesses elf ene ltind er anether usually in nite series Many special equatinns at parstieular irnperitanee in applieatiens fer irnstanee these et Legendre and Bessel rnentienedd in Seelien l have been studied at great length ainti the ll eClIquot let a sinle sueh equatien has eften been teuind set eempilieatecl as tel eertstittite by itself an entire rlepartment ef Eil1Ell EiS We shall tliseuss these niattiers in Chapters 5 and In this ehaplteri ear detaile eensiclleratien ef aetirlatl n1etheds fer selsring wiltli he restrieteel f lvf the mast part In the special ease in whieh the eee ieienlzs Flat Ell39Illl Q x are eenlstanttsi it slietlld arise be ernphasiaed that must ef the ideas and preeeszlutes we drisetrss earl he generaliaed at enee tel linear eqnatiens ef higher Fdef with ner ehange in the underlying prineipl es btit enly an inereasing ieempltesity in the surrellnding cletaiilsi By erestrrietlneg eurselires fer the nrest part tn seeend ernler eqllatletns wet attain as mueh simtplieit39y39 as peissilhlle nritherut tlistertirlg the main iizleas antil yet we still hiaviel eneugh g39entetalitt39y tn inetlude all the linear equatins at greatest interest in rnathematies and phi FSIE es 0 Sinee in general it is net pessihle ate preduee an iersplieit selutien at fer inspelti en ear rst erdlerl ef business is ate assure eurselvres that tl 1is equatien reallquoty has a iselustien The fellewing ienistetneie and unique ness theetrerrr is pIquotl39flI lrquot3 in Chapter 13 Thaearem E Let Pit Qlaz and Ring be eent1inaeas ferretiens en a nzlnrseel tiater es39l lunhquot if In is am paint in lah and if ya and ya are any eamlrers39 rshnrlsirrer than tT jquotlt Iti Jfl 1 has ens nirral wily ans seilttrtern hylfx ea rile entire inrernel site that yn y e39rltrd 39wrl y if n and it are seal ntzlntbers stleh that e lr then the sf5rrnlrel llngbl deneles the inite39nral eensisting at all real nnn1rherss that saitisty the ineqnalilises tr 5 I 2 la quotl h39is inlervarl is ealleri elrrsel heeatrse it eentains its enep einls The eyes inlersral reselling treat the eseluslen at the entlpeinls is tlenntetl by al39a anti is e uneel lay the ir1eqnalitiles is I s la SECOND ORDER LINEAR EQUATIONS 83 Thus under these hypotheses at any given point x0 in ab we can arbitrarily prescribe the values of yx and y x and there will then exist precisely one solution of 1 on ab that assumes the prescribed values at the given point or more geometrically 1 has a unique solution on ab that passes through a speci ed point xy with a speci ed slope y In our general discussions through the remainder of this chapter we shall always assume without necessarily saying so explicitly that the hypotheses of Theorem A are satis ed Example 1 Find the solution of the initial value problem y y 0 y0 O and y O 2 1 We know that y sin x y cosx and more generally ya csinx C2 cosx for any constants c and C2 are all solutions of the differential equation Also y sinx clearly satis es the initial conditions because sin0 O and cos0 1 By Theorem A y sinx is the only solution of the given initial value problem and is therefore completely characterized as a function by this problem In just the same way the function y cosx is easily seen to be a solution and therefore the only solution of the corresponding initial value problem y y 0 yO 1 and y39O 0 Since all of trigonometry can be regarded as the development of the properties of these two functions it follows that all of trigonometry is contained by implication as the acorn contains the oak tree within the two initial value problems stated above We shall examine this remarkable idea in greater detail in Chapter 4 We emphasize again that in Theorem A the initial conditions that determine a unique solution of equation 1 are conditions on the value of the solution and its rst derjyative at a single xed point 20 in the interval ab In contrast to this the problem of nding a solution of equation 1 that satis es conditions of the form yx yo and yx y where x0 and x1 are different points in the interval is not covered by Theorem A Problems of this kind are called boundary value problems and are discussed in Chapter 7 The term Rx in equation 1 is isolated from the others and written on the right because it does not contain the dependent variable y or any of its derivatives If Rx is identically zero then 1 reduces to the homogeneous equation y Pxy Qxy 0 2 This traditional use of the word homogeneous should not be confused with the equally traditional but totally different use given in Section 7 If Rx is not identically zero then 1 is said to be nonhomogeneous I1I IFTFEIt39E tJTii tL E5U39UATt HE In sttud ng th no nh0m0gen e u5 Eqtlatim 1 it is nacxassary tn mrnsti der ailmg with it the hntmtngen mu5 eqtuattttitnn Dbt itt dt tmm p by rapImit1g R by H Under these Citctutmtstanctzs 1 is Dften 39C E d the tJ IpEIE39 q39 af I39i FI attdt 2 th r tduc ti equarttm 3550CiatEd witth P Th jt E S ftiuri Iit39ttC gE bi twt 1 and p is easy ma understamtdt as f fInw5 SllppDEE that in same way we know that ygxc1tc is the genetratlt 311hltitzttt ttf p expatct it tn Et1rttai1t tw arbittary rDnS ttantS since th equatian is at the smatnd l39dE139g tftd that ypzft S ta tt d p EquottiiE39U F tstutti n tilt L If ytx is any sttttttiunt whatever of 1 l lft an eas3r C lEtlJ ti 39It thaws that yt xte ypx xjt is O S l1t Il of 2 9 E X 4 PtErty yt QfxmI M 4 Pt y M yti Ptxtltat RE 51 Di 3 Sittte y xttct w is tI he gE139Itr39 l totutitzrn of 2 it tf tlltuwts that ytr 4 pI yl l tI k rll tt M Nit JtIum392 mm rm at suiIabEe ctthnim at the t3Utquotlt5 t t139IS tit and 2 This tatttg11ment pmves 39t 39IE tf3nlnwitrtg tihtemru3tmtt ThEurem X I39fyR tit Hmr3 ger1 raf tnt39ttt t rt 0 th39 rat umtd Eq QIE 0 ttnd 8 tit may tar ttutTttr 3e3tut tn ttf tthg tittntppfetret t3qttttt39 m I the2tn y 4 Jtr E3 the ganttra39quot 5quota2tt39tutrttm turf 1 0 tsha t age in Sectiimnz 19 that if yg is krt wtti then a fmm l iprmed ut39c is tlvaitlabt far ndi11g RFD This 5ahD 39w395 that the tttzrttral Vprtnmem in th Th E Difquot of linear equatiDnS is that at sulving 39lLhE Lh TtDgEi E IUS equa39ti mt AEE rdingl39 mast ur attentti n will be dtettv39 Dted ta stttdyitng the E1 I39t1 Ct l1 1quottF39 t lf yg and irtvetigattng vatrtimlxs ways if Lttttrmining its seiatplitiizt f39trmarnrJnra of which is tffKt3EIi quot39EI in all G 5E5 first thing we stmufld HQticE abtmtt that htmmtgEt1mwut3 q uatiD n its tlhat 39l39EltTt functi n y1 twthi h is idEfnl iEt t 39y zt39rtrmthat is H1Jj C it all r is always a 30tt1tttrnt This is ralfIett the fa39Iquot 39tIt39ttI 3v tTtIl iAt3f and is usruatiity Of no ii 39lEt E IIt The b i 339t139ttcturaiZ fact aab ut 5t ltutttiu1nst f 2 is gtivven in the foilowitng tthtent1rgtetn1t Theuremt C if Jli3l and ygufxit HFE ttny 39Wtt ttu ur iGrLt 1 if 2 must 3IiJ I39 C 52323 P tsquot m 39m tr 5tJftm39tm far any t 39r1i 39 rIt 393 attcd E3 SECOND ORDER LINEAR EQUATIONS 85 Proof The statement follows immediately from the fact that 51 Czyzquot PxC1Y1 C2 239 439 QxC1 1 CZY2 cry czy Pxcyi czy Qxcny1 Czyz Cilyl Pxyi Qxy1l Czly Pxy Qxy2 c0c2O0 5 where the multipliers of c and C2 are zero because by assumption y and y are solutions of 2 For reasons connected with the elementary algebra of vectors the solution 4 is commonly called a linear combination of the solutions yx and y2x If we use this terminology Theorem C can be restated as follows any linear combination of two solutions of the homogeneous equation 2 is also a solution Suppose that by some means or other we have managed to nd two solutions of equation 2 Then this theorem provides us with another which involves two arbitrary constants and which therefore may be the general solution of 2 There is one dif culty if either y or yz is a constant multiple of the other say y kyz then C1 1 C2 2 239 C1 Y2 C2 2 C1C C2 2 CY2 and only one essential constant is present On this basis we have reasonable grounds for hoping that if neither y nor y2 is a constant multiple of the other then C1 1x C2 2x will be the general solution of 2 We shall prove this in the next section Occasionally the special form of a linear equation enables us to nd simple particular solutions by inspection or by experimenting with power exponential or trigonometric functions Example 2 Solve yll y 039 X By inspection we see that y 1 and y e are solutions It is obvious that neither function is a constant multiple of the other so assuming the theorem stated above but not yet proved we conclude that y c cze is the general solution Example 3 Solve x2y Zxy 2y O D39I FE EREI iTIAL EtLTAT39IU hi5 Since di erentistirtg is pewer pushes tliewnt the mcpenent by enes unit the term it this eq t1atien suggests that we teeth tier pessihile se1ittieIts ef the type J Ai quott Uh st1hstituting this in the tii er39entis39l ettatmn end dfisriiding hy the etirtnmeh teeter t we ehtein the qu1t1rettie equetien J39IlJ 1 1 2n q D et 11 l r1 4q D This hes resets ti 2 1 21 see ya J ems 312 12 are seiutttiens end y in e39239quot2 is the gesriersl seh1tieIt en any intieWel quotnet eeriteihing the erigit1 It is wrmrth remarking st this p i39l39 tIi that e late part if the sth eet39 Elf ltirteat equetitims tests 011 the fl3i1l TId39lIquotI quot39llEal39Il1fEIi pmperties started in Theorems B tted Na Ah inspeetietht of the C lth3921t1i ti S shed will shew st mice that these preperties in tum depend on the iiheer ty ef d e rentiietie nt that E5 on the feet that i ftstj s39is39 f E 4 s39ri fer all ennstsrtts his end B titfi all cli 39erehtiehie ftinetiens ffxi and gtrx PRUBJILEMS tn the feilewihg ptehtetms essuzme the quotfeet stated shewe hut net yet pvreverit that if 0 X and K3ii ere twe sehitiens hf V and neithei39 is s eetistant n1uitipi et ef the ether then e1ys eEyEIs is the gie139iere sehttietm Q F 31 Uetriify that Vy 1 and y 3 are seltttiehs ef the redtzeed equstietn y 3 I and write dewh the gehetai S U39Ii 1 pe Determine the veltte Df it fer which ME ex is e per39tieLiist SDitliitlltt ht the etemplete l3i39L1 ttiitUiT39l J 3939quotquot 353 Use this seli1t39ie139i erid the result hf pert en te write dewn the tgenerel selittieh ef this equettieh Cerhpste with Etemple 1 in Seet39ien Til 0A CitHt yeiu CiiECDquott quotEI y y39 and yp by if tS3Iti39t31Ii1 t1 s39erity thet ya It and yz Iegx ere selutiiens emf the eqhetieh 39yquotquot yquot Eh end write dewh the gesnezrel Eat3itItttZlt39t Can 39U tt ICiiLSE quotv39ET yi and gs y i nspeet iten Ti ifs Sheer that y e end y2 e ste seiutiens ef the rediiieed Eq iti y y Zy g What is the genete seiutient 4 Find 1 E1391quot39Jd h sh that yp ttx 5 is e pertieiJIstseit1tien ef the eempiet e lJJEtitCI 1 y yquot 23 41 Use this selutieh and the resuit et pert 3 to write kdDwn the general SDiU i t39l hfquot this eq Listivrn Use insheetien seer experimient the nti Kat Z quotEh391Ji t39 seltitien tel eseh ef the feiiewintg eaq39u1s t ietits at y sly is 1 hi Iquot 2s ta 2 y Ly sin st in eeeh est the feilhwih eases use irhspeetieh et es petrimeht te nd patitit tli t St iit LtIitnE ef the 39l39E 39LtEEti and eerhp1e te evq1istiehs arid writte tZi W 39t the getietsi 10 15 87 SECOND ORDER LINEAR EQUATlONS solution 3 Yquot 6 b Vquot 2y 4 C yquot y sinx By eliminating the constants c and C2 nd the differential equation of each of the following families of curves d x 1yquot W y 0 e y 2y 6e a y clx czxz e y cx c2 sinx b y clef c2equotquotquot f y cequot czxe c y cs1nkx czcoskx g y cequot c2e 3quot d y c c2e 2quot h y cx czxquot Verify that y cx c2x5 is a solution of xzy 3xy 5y 0 on any interval ab that does not contain the origin If x 0 and if yo and y are arbitrary show directly that c and c2 can be chosen in one and only one way so that yxo yo and y x0 y Show that y x2 sinx and y O are both solutions of xzyquot 4xy x2 6y 0 and that both satisfy the conditions y0 O and y 0 0 Does this contradict Theorem A If not why not If a solution of equation 2 on an interval ab is tangent to the xaxis at any point of this interval then it must be identically zero Why If yx and y2x are two solutions of equation 2 on an interval ab and have a common zero in this interval show that one is a constant multiple of the other Recall that a point x0 is said to be a zero of a function f x if fixo 0 THE GENERAL SOLUTION OF THE HOMOGENEOUS EQUATION If two functions fx and gx are de ned on an interval ab and have the property that one is a constant multiple of the other then they are said to be linearly dependent on ab Otherwise that is if neither is a constant multiple of the other they are called linearly independent It is worth noting that if fx is identically zero then fx and gx are linearly dependent for every function gx since fx O gx Our purpose in this section is to prove the following theorem Theorem A Let yx and y2x be linearly independent solutions of the homogeneous equation Yquot Pxy39 Qxy 0 1 on the interval ab Then Ci 1x C2 2x 2 SECOND ORDER LINEAR EQUATIONS 89 Proof We begin by observing that W yiy f yiy yzyi y yi yy yzyi Next since y and y2 are both solutions of 1 we have ya Pyi Qy 0 and y Py Qy20 On multiplying the rst of these equations by y2 and the second by y and subtracting the rst from the second we obtain yy 3 yzy Pyiy yzyi 0 dW PW 0 dx The general solution of this rst order equation is W ce39 quot O and since the exponential factor is never zero we see that W is identically zero if the constant c O and never zero if c at O and the proof is completes This result reduces our overall task of proving the theorem to that of showing that the Wronskian of any two linearly independent solutions A of 1 is not identically zero We accomplish this in our next lemma which actually yields a bit more than is needed Lemma 2 Ifyx and y2x are two solutions of equation 1 on ab then they are linearly dependent on this interval if and only if their Wronskian Wyy2 yy yzyf is identically zero Proof We begin by assuming that y and y2 are linearly dependent and we show as a consequence of this that yy yzyi 0 First if either function is identically zero on ab then the conclusion is clear We may therefore assume without loss of generality that neither is identically zero and it follows from this and their linear dependence that each is a constant multiple of the other Accordingly we have y2 cy for some constant C so y cyf These equations enable us to write yiy yzyi yicy cydyi 0 which proves this half of the lemma 3Formula 3 is due to the great Norwegian mathematician Niels Henrik Abel see Appendix B in Chapter 9 and is called Abel s formula t t mEeeeeHTmL EmampT t t45 We new asmme that the Wrenektant is t rtentieaItv eere anti preelvet Iiineer tcfe39peendeneet If yi is identieeIly eer en then fee we iremettted at the beginning et the seetim1 the ftunetiens ere tjznteerljg depletfldenjt We may theret fm e aseume that y ees net ea39nieh ietentiea1iy39 en ebi 5 whie11 it 39featIJ39w e 0 eet1ti11Ltity tthet y detee neat vanish at all en eevrnte et1bin39tetrvat ed of 935 Since the WItD 5Ei is itclentiea1Ttjy mere en ame we eatquotJ d i IA39it ee it by 0 t get 3939Jr39 Eye yt 0 en edt This E t be wrirttetn in me term y2fyl D and by integrating we erbtein hjy er ygx e ky fer sesrne etenetant A and at J in etdit Fina tt1y einee y3e and kVye litatvtet equal values in ed they have equal dereiieativetst there ee wejtt and Tttheerem MA allewe us te infer that J e 3395 E kfmixl fee at x in e51 wtiteh veene39t39udee the ergurnentt With this lemme the preef emf Theeeem is eempllte te Ordinarily the eimptteett wa39 P sheewitntg tthat two setitutitetne ef U are tttineerlyr tindepeemlent mete an interval is tee shew that tzheisr mtie net eenstant tt1eret anstrzl in rneete eaeee PAK is eaeitgr dteterminedt by im5peetitmL On Ua C 5ivtJi39t hewetver it is eentvetnient tee emtey the feretmat test embedied in Lemma 2 eemtpme the Wtrenekient and shew that it cjttjee mt vani gth Bth prv Dd39LtrES are tl1tetrett etdt in the f lttewing etxemptet E te39mple 1 Shew that y e sin 1 e eere I is the general eetutttetn if yquot y 2 0 en any intt enrat1 arid End the partieuler St I39LtIiiDt lJ fer wL1iet1 ylf T and yquotw 2 Thee tiaet the y E it LJ and 313 E l391DE1 are eututine Ee easily vent etdl 113 e11itaetit1u39ttt e I1t 39Their tinear intdeper1d entete on ten iKnter39vat at tettews fzrem the ebetervatittetn that fTFquot2 remit is net eemsten t andi else 39frnm the feet that their 39W r39enetden never vatniehesz eiinI eU5 2 N quot2 5m 1 em I 1 W l E quotFl hi teem quotStir x 5ii tee Pfx U and 1 are e nstin1ee usUn it ntrmw fevltlewe fF m1 Theerem A tttmt y e sinr 1 eeetxt ie ttate ten earal3 selIJttten ef me iigtvetn Eq1JtitEiU mi FUrtihermere stnree the t39mtervel G 13111 be ex pended imet nittely withtJrut intmdueieg pe ittte at which P 3 1511 QIQE is eli5ee1n tirtmuust thia gteeeeet sevtutie n valid tier ell xi J es nd the required t tquotquottiEL1ittIquot seluttient wet solve the eyettetnt e eeuei E3 Sin 3 2 yijetds egg 2 and e 3 ea y E Eetne Eteeer is the partieu1ar eejltutien that eatie tee the gtvezn eeen ditieene 91 SECOND ORDER LINEAR EQUATIONS The concepts of linear dependence and independence are signi cant in a much wider context than appears here As the reader is perhaps already aware the important branch of mathematics known as linear algebra is in essence little more than an abstract treatment of these concepts with many applications to algebra geometry and analysis PROBLEMS In Problems 1 to 7 use Wronskians to establish linear independence 1 2 Show that e and equotquot are linearly independent solutions of y y O on any interval Show that y cx czxz is the general solution of xzyquot Zxy 2y O on any interval not containing 0 and nd the particular solution for which y1 3 and y 1 5 Show that y ce czez is the general solution of y e3y 2y0 on any interval and nd the particular solution for which y0 1 and y 0 1 a Show that y ce2quot czxez is the general solution of y 4y 4yO on any interval By inspection or experiment nd two linearly independent solutions of xzy 2y 0 on the interval 12 and determine the particular solution satisfying the initial conditions y1 1 y 1 8 In each of the following verify that the functions yx and y2x are linearly independent solutions of the given differential equation on the interval 02 and nd the solution satisfying thestated initial conditions a y y 2y O y equot and y2 e 2quot y0 8 and y390 2 b y y 2y 0 y e and y2 e 2 y1 0 and y l 0 c y 5y 6y 0 y 63392quot and y2 3 yO 1 and y 0 1 d yquot y O y land y2 equot y2 0 and y 2 e 2 a Use one or both of the methods described in Section 11 to nd all solutions of y y 2 0 b Verify that y 1 and yz logx are linearly independent solutions of the equation in part a on any interval to the right of the origin Is y c c2 logx the general solution If not why not Use the Wronskian to prove that two solutions of the homogeneous equation 1 on an interval ab are linearly dependent if a they have a common zero x0 in the interval Problem 1410 b they have maxima or minima at the same point xo in the interval 9392 DIFF mHTIA1ELm TtnHs Ctmn5idEr39 the twn funzztinns 3 35 and gv m3x an iilh iI39ItET 39s393 L IL 3 Shaw tihat th ir Wrbr15kian f g vanislhes iexnti ralr Lb i5h w tgihat f and g are Armt l39izJ7uearIjr Llij p s dEn rcjc DE 3 and ism wnt1radic39l LEmma 239 P u rmL why mm 10 It i5 cigar that sign I EDSJ and 54in I auinx war are twin distincst pairs inf linearZEj39 indJependent sulutioans of yquot e 0 Thus if and are linear ly39 inlz pendent 5ul39ut39iiu1n5 Inf the hDrn t1gEI1Em39u5 Eq39uatinn wPmwum we see that ym amzl ya Aare nu uniquely datermined Eh tEqu a1 tiampDn S39hIZIW Eh l x quotI 2 F PH T WU39nJ J HI and V Vyiiy VJa39 y39 Qixl W Um M an that the equatiun 5 uniquiy39 determ4ined 39bj39 any given pair if linaaariy independcn t E l ti 5 n 11 Use 13 to rejcuim5trrm7t tiilme Equa li n F 3 A U fmm Each Bf me 0 p E39iTE D E IihnearfjF imVdV ependEntA 5 uti n5 1EIIITi7D E39Ij a39bm39e 2 USE H31 rm reTunrn5truut the equation in Pmblem 4 fmrn the p air tjf jlinear 3y i ndep enndent s lutgiuns 3139 Al gh Ii Show tmt by apr Fing the 5ubStfitufim n y M1 to the hrawmng n us Eq mijtimn 1 it is pm55ih1Elt mt Cbt in 1 Ifm mC1gEn5cIu5 sec nd gDrdEIT linear q1m31ti n far 1 wit391 nu LN term ZlT E5E l39l39L Find 1 and t1vEequatim1 f r LI in terms of the nriginai cce ii En39t5 F x and Q xaV bl Use the rm1Et hc1ri mf part 3 In nd the neat sulutimm Elf 0 214 xi P P P As we have EEEi1 is easy lE m wrim dmwr1 the general s EVutinn umf the hUTTT gE f MS cq11a Iim1 PMW QUMU H WHIGIHEVEIT we ki1 quotW39 KWD Ii B 2 39 indc p B ndm1t 5Ulu ti n5 J amI Em how 117 we Ky and yg U mf rlunatEthr 39th em is im gEn raE mmhnd far dming H nwewar thare ampLI u es exist a standard prDredure fur dampIerminVing wl1en ya is krmwn This i5 maf cmn5id rahIe imp0rtanrE3 far in nmny ua5e5 a 5ingle Emlutiu4n 0f 1 can be found by i S pE C tiU at smm GthEI devi E To devBlUp this prrunedu39rt wt a35umAelt quotthat yibz E1 kn wn nanzeru sUluIiun Uf 1 so that yx is alsm P 5u1mMiun I rr any mnsIan t SECOND ORDER LINEAR EQUATIONS 93 c The basic idea is to replace the constant c by an unknown function vx and then to attempt to determine v in such a manner that y2 vy will be a solution of 1 It isn t at all clear in advance that this approach will work but it does To see how we might think of trying it recall that the linear independence of the two solutions y1 and y2 requires that the ratio yzyl must be a nonconstant function of x say u and if we can nd u then since we know y1 we have y and our problem is solved We assume then that yz vy is a solution of 1 so that yZ Py QY2 0 2 and we try to discover the unknown function vx On substituting y2 vy and the expressions y vyi v y1 and y vya 2v y v y1 into 2 and rearranging we get Vi 439 Pyi QY1 Uquot 1 U392 i Pyi 0 Since y is a solution of 1 this reduces to v y1 v 2y Pyl 0 OI saR Y1 An integration now gives logv 2logy1 fPdx so Ur 513e fPd Y1 and 1 Pdx v 26 dx 3 Y1 All that remains is to show that yl and y2 Uyi where v is given by 3 actually are linearly independent as claimed and this we leave to the reader in Problem 14 4 Formula 3 is due to the eminent French mathematician Joseph Liouville see the note at the end of Section 43 A94 hF F39FERENT39IFhL E LJsquotkTjDHE Eimmple yr 1 is a reILIlien ef hearquot 3quot w y p whieh is simple eeneugh e he dieeverrehd by inspection Find the genereelle eSI 1luJifiU1n we begin her 39erming the men ewmeetiehh in the ferm ef 1 J 1 13 quotJ 1 Jrquot 4 f 3 I I Sitnee P 139rK e eeeeemi ampieneeeri1 indepener1t eee hut39i e n 155 given 39hr ya uyl where 25 UJ EE l39lrdDfxJEIg J I This yieleleeyg 8 if 1 2139 e rise the genereI eeluteievn 8 y e hei1requotquote PRO mi 2 Irf ym a neeneere eelutien ef hequatien 1 and y eyh where 1 is given hy fermule 3 the eeeend eeleutien feeend in the eteext shew he eempetihg the hWre neefkien that end z ere Hineer Iy indveependem Use the methDd of sseetien the nd 5 amzl the genere1 eelmieen ef eeeih ef the IDHIIUEFEMEE eqeeiiene flreim the given selL1tiIifrm 3 aI 3quot Jr 0 J 0b9 lib 9quot P V J 3 E The eqeetieen Jry 3y t D has the ehrvixeue ee leut ien Ayah 1 Iquotigned yr end the general sILu39tium Iv that y II ie one e iutieh ef rayquot hey 43 U end nd ya and the egenerel eellmien The equetieen 1 E 1quoty 39 Ely U is the epeeiefl ease ef Legendrequot5 eque em 1 I3 p Pie 139 039 eerrheepending te p 1 It ehsee yl r as an ehwi eee eelueteielh Find the gen ereli SDIMtiCI3911 0 eque iirrr1 M ey39 I3 ijlly 2 G is the eJeuie teee eaf Eeeee e eque en W III p1v39 U eeerereependirng me p K VE1li ifquotV tihert y pB 3 meien eh ene se iutien if f eny imereei ineh1eedin g enly epehj1iwhhe eehmee ef 1 and heed the genemle E lM LE UnU Use the feet thheirt y E u an e hvjeus eeluhen evf emth ef the futlelewieng eqmJeeien5 Ce find tiheeir general eelJut iene s e H e 1 W N he ya 0 1 1 3 1 p r 2p Yq p 39 p q lt V e gray rm 1 2ye r39 Ely Find the geenerel eealutiene ef y xfxy Ixy If V7eri39fhy that ene eUmtie h f Eyequot P 6 lily 123 11 U is given 39lI39ajr39 5 e d nd Itfhe genereel e mi ren SECOND ORDER LINEAR EQUATIONS 10 a If n is a positive integer nd two linearly independent solutions of xy x ny ny 0 b Find the general solution of the equation in part a for the cases n 1 2 3 11 Find the general solution of y fxyquot fx ly 0 12 For another faster approach to formula 3 show that v yzy Wyy2yf and use Abel s formula in Section 15 to obtain 1 17 THE HOMOGENEOUS EQUATION WITH CONSTANT COEFFICIENTS We are now in a position to give a complete discussion of the homogeneous equation y Pxy39 Qxy 0 for the special case in which Px and Qx are constants p and q y W qy 0 1 Our starting point is the fact that the exponential function e has the property that its derivatives are all constant multiples of the function itself This leads us to consider y 6quot 2 as a possible solution for 1 if the constant m is suitably chosen Since y me quotquot and y m2e quotquot substitution in 1 yields m pm qe O 3 and since e quot is never zero 3 holds if and only if m satis es the auxiliary equation m2pmq0 4 The two roots m and m2 of this equation that is the values of m for which 2 is a solution of 1 are given by the quadratic formula Ip i V192 4 461 2 5 rnl rn2 Further development of this situation requires separate treatment of the three possibilities inherent in 5 Distinct real roots It is clear that the roots ml and mg are distinct real numbers if and only ifpz 4q gt O In this case we get the two solutions f zx equotquotquot and 6 Since the ratio DI K E015 A Tl39UH3 is HUI c mstant these smluti0ns are Ii nEarly independgnt sand is the genEraI 5 IUti U4 of 1 Distinurt cnmpl ex rnVots5 The r L s ml and mg are distirut tznmpi zL numbers if and 1 139 if pi 4q 1 In this case rm and mg can be written in the fmm 1 i Eb and by Em39er3fu4rmu a E 2 CD5 A 7 sin 9 T D1121 mm siuiimias at 1 are em M 5 W e r3 quotquot c 5 bx 1sii1 br BM and Em3 e Em I 4 e equot EMCCDS bx iv fsiin p M Sinc we are imere5t ed mniy in snEutrimrns that are rampeavaIued fun ti ns3 Vwe Lian ad5 C8 and 9 and di Vid E by P u and subIracI arul dividg by 21 to Dbtais n kg 05 bx and E sin bu39 I0 Thsase mlutmns iam linearly inda p Emdent an iifh g ncrral SE lU Iin of 1 in 1his CHEE is y E39u vE1CDS I539I C2 sin bI 11 WE EH11 I k at this matter fmn anmhmzzr p int of vieiwy A C m plEK v u d functiml wx u l1I iur aajtis es equatisan 1 in Vwhich p and q are rm numbEr and Duly if 1 I vx satisfy segpargatrehr j5xcmrdingJiy Aa mpl ampx s 1iuti1un Df 11 aV1 way5 mntains twu real s mIVutim15 and 8 jyiel d5 the twn E jIl139ETiU S V161 mince Equa real r 39ILs It is Evi nt that the mats rm and W are mzlual real ru11mberS if aaand July if p2 4qi 1 HBlquotE WE htain i1nlquot UHE s lutin y E with F B HoVwewr we can easily G 3 5BCUii lJ linyeamly im4leendent 5 JIutim1 by the T 39IBth d 5I the preceding sacti1rn if WE take C e3quotquot Jquotm 39 39 th ern P 7 2 39rFi dI J irt E If ya 9 RWE EaU it Fm gIan Lrd thai tha readcr39 is E qU iI139l 39Ed xe39Eth 39Ihe1 EltIli11eI1tar39 l bf af cmmpe x slmnrzbczrs E LIE1quotS mrmuia F is em zmght m bv a smnard jpart nf anjr l39 39iEJI1v lZIhvquot 5a39E ifaamr39 C LllquotE in caluhnE SECOND ORDER LINEAR EQUATIONS 97 and y2 vyl xe quot In this case 1 has Czxemx j quotIX yC1e as its general solution In summary we have three possible forms given by formulas 6 11 and 12 for the general solution of the homogeneous equation 1 with constant coefficients depending on the nature of the roots m and quot12 of the auxiliary equation 4 It is clear that the qualitative nature of this general solution is fully determined by the signs and relative magnitudes of the coefficients p and q and can be radically changed by altering their numerical values This matter is important for physicists concerned with the detailed analysis of mechanical systems or electric circuits described by equations of the form 1 For instance if p2 lt 4q the graph of the solution is a wave whose amplitude increases or decreases exponentially according as p is negative or positive This statement and others like it are obvious consequences of the above discussion and are given exhaustive treatment in books dealing more fully with the elementary physical applications of differential equations The ideas of this section are primarily due to Euler A brief sketch of a few of the many achievements of this great scienti c genius is given in Appendix A PROBLEMS 1 Find the general solution of each of the following equations a yquot y 6y 0 139 yquot 6y 25y 0 b yquot 2y y 0 k 4y 20y 25y 0 C yquot 8y 0 1 yquot 2y 3y 0 d 2y 4y 8y 0 m y 4y 6 yquot 4y 4y 0 H 4yquot 8y 7y 0 f y 9y 20y 0 0 2yquot y y 0 g 2y 2y 3y 0 P 16y 8y y 0 h 4y 12y 9y 0 q vquot 4y 5y 0 i y y 0 r y 4y 5y 0 2 Find the solutions of the following initial value problems a yquot 5y 6y 0 yo e2 and M1 see b y 6y 5y 0 y0 3 and y 11 D yquot 6y 9y 0 y0 0 and y 0 5 3 y 4y 5y 0 y0 1 andy 0 0 e y 4y 2y O yO 1 and y 0 2 3i f y 8y 9y O y1 2 and y 1 0 3 Show that the general solution of equation 1 approaches 0 as x gt 00 if and only if p and q are both positive 4 Without using the formulas obtained in this section show that the derivative of any solution of equation 1 is also a solution 93 DIF39FEEE1HTiAIaEQ1J39 TEITJWE quot 5 W aquatia n E wmtra p and q atra ar1natanta iEEr39E aIEEi Eaa arquotr5raq39ufd ma n3im1a aaqwuaLE mn Sh rr w that the aihanga af indapaanulan39t var39iab la Agiaan by I E tra11afarmaa it ianrta an aqaati n wiltLh aanatant aava manataa and apply this 1aa1aiqaa ta snali ma gaaara aalutriarn atf E h t1f39Il1aa f al laawiang aquatians aIw 1r2r quot39 ID D f x2y39quot quotlayquot y r fa taxEy39 1 xy B 4 0 a Zrquot39 33 4 D E IW A 13 3911 E3 3 4 arj 23 0 H 4a3yquot39 E 33 E IIBI 2 yquot xy l y A1 W Ila 3xJ 9 D In Prabalaann 0g3 cart39ain hamagananas aqaatiatna with vaariaLab1a 39rrcmai anta were traanafa rm ad imxa taquaataiana with canatanst aaa iaianta bgyr rIIangmg the imla pnnd an1t vaaarialala from I to 2 2 Iaga Chaansia ar Elma garraarail hDmDgEnE ua aq1aaatia n 9 Wily Qxa U W amiJ chana tha Lin apandant vyariaabita fraam 1 to a ax wiLhafra ax is an unapazci ad fuanatiaan af x Shsaw that aquat iavn 3 can lama traansfarmasda in thzia way Ema an aqaaaztiana with cannattantgf aianta if and anl P if Q 2PQftQW is canataat in which case a S I wm a am tha daaairada 1raauI39t m the raaulat at Paraahlarn 6 to disaaavar w39harthar aaaah the fa1 a39winag qua r395 can be tranaafarmadquot iata an aqaatmza wai n am natam aa iaiams 0 hangir1g the aiandependanta raariabEa and aaalvaaa it if P 39lquot5Sib1Ei a 1yquot39 raE r 0 x 0 0 E 3quot 3vrquotquot gag U9 In this 39prsa39bIam wa praaan39m an0Ehar Way of dialaawaring tha aac rn liaaaraly indiepa39nd1ant E 39ElJtirD af 1 Wham tha rmzwta at the aauas isaw aaqamaaatiinn arrpa real and aqI1aL a If mat my waarify athast tfha adiffar39Enatiai aquaataian N FE 1 3939 E mamay Ahas V EquotJquot39 52 EPHVEJE P mm mg as a aawatiana 1fb GI mg as 39 xaa and use H39a5piEtaal 39a ruia a0lutinn in paggrt P aa rm r EH3 fa Verify that the limit in part Eb aatia as tha Vzliffarantiaal a qaatiaa n ahatiainaad farara ma aquatimru in part a 0B reil1acirng m by rngz ta mrld the hmirt ad the a It ahaa Imsnjwn aa ffaua 3quot 3 aafuicfjfrrwrzaatarmai aquaria39a Euaiaaar a rasaarrhaa ware 5as aI En5i 39FE that mam rnaEhaInatEaiana try la aamrniid aanamaian hy naming aq ua im1aa fwarm uIa5 thaarama ala far tiha paraaa whpa rai aIu dijad tham aftar Eunlarr SECOND ORDER LINEAR EQUATIONS 99 18 THE METHOD OF UNDETERMINED COEFFICIENTS In the preceding two sections we considered several ways of nding the general solution of the homogeneous equation Yquot PXy Qxy 0 1 As we saw these methods are effective in only a few special cases when the coef cients Px and Qx are constants and when they are not constants but are still simple enough to enable us to discover one nonzero solution by inspection Fortunately these categories are suf ciently broad to cover a number of signi cant applications However it should be clearly understood that many homogeneous equations of great impor tance in mathematics and physics are beyond the reach of these procedures and can only be solved by the method of power series developed in Chapter 5 In this and the next section we turn to the problem of solving the nonhomogeneousequation L y Pxy Qxy RX 2 for those cases in which the general solution ygx of the corresponding homogeneous equation 1 is already known By Theorem 14B if yx is any particular solution of 2 then yx ygx ypx is the general solution of 2 But how do we nd yp This is the practical problem that we now consider The method of undetermined coefficients is a procedure for nding yp when 2 has the form yquot W qy Rx 3 where p and q are constants and Rx is an exponential a sine or cosine a polynomial or some combination of such functions As an example we study the equation w y py qy 6 4 Since differentiating an exponential such as ea merely reproduces the function with a possible change in the numerical coef cient it is natural to guess that yp A6 5 might be a particular solution of 4 Here A is the undetermined coef cient that we want to determine in such a way that 5 will actually DtFFEIt EHTIAL EmuAT1mts S tiSfy 4 On t3 ubstituting 5 intm 4 we get AaE pa qE t EM 30 I H p I q 6 This vallue at A wilt make 3 s ltuti nt Hf Heep when the dE miH I39 T cm tits right 0 6 p quotEt390 The s urce of this Lii itcultty is easy In Ltnd rsttand fmquot the E1tquot39IEEp It arises when 1 is 3 mat of the auxiilittaty equa n mg 39 pm q t U and in this cast we mow 5 reduces th left side mt 4 t0 zero and cannt pmssi39baIy tsatizsfy 4 as it stands with the right side di tertent fmm EEITEJ What can be done to 39 J tIiI39IILElE the tproceduret in this E21tEE39ptitt2tt11AI ctts We saw in the prmri us s ctinn that twhetn the atuxi1itatr etquattiun has 3 duble mot HIE S d linearI tindependcnt solution at the hE39f DgfE tnsans Bqittati n is ttbtaitnetti by muI5tiptlring by Jr With tltzis as H hTitl39IL we take yp Axett 8 as a stuttbstitute trial sotlution On inserting E into ft we get AtE tr pa qIt E 4 AICZH 14 pj lt quot The 39 IquotS1I et rtessitmnt in parentheses is zem betca39usE uf mlr assumtpticrn that 3 is 3 must If 7 30 It A 39 L Ea p tt93 This gives 3 vagigid CUE f C39iE t fm SJ texc pti when H p2 thtatt its EtCEP39t Wham it is a d uble meat at In this scam we hopefully E th 5 uct1e3sful tpatttern i di t d abmr agttd ttry yp Atria Suhstttutitz1391 tftf P into yields pa qx Ee 39 E pj1re quot ZAEM e Since I1 is Iii W SS1IIquotl 1E39 I10 be H rdmutbttet mm Ulf IquotE39 bmht Ettpre5sitDn5 in ipaarenthesas yam E lquotG and 11 SECOND ORDER LINEAR EQUATIONS 101 To summarize If a is not a root of the auxiliary equation 7 then 4 has a particular solution of the form Ae if a is a simple root of 7 then 4 has no solution of the form Ae but does have one of the form Axe and if a is a double root then 4 has no solution of the form Axe but does have one of the form Axze In each case we have given a formula for A but only for the purpose of clarifying the reasons behind the events In practice it is easier to nd A by direct substitution in the equation at hand Another important case where the method of undetermined coefficients can be applied is that in which the right side of equation 4 is replaced by sin bx B yquot py qy sin bx 12 Since the derivatives of sin bx are constant multiples of sin bx and cos bx we take a trial solution of the form yp A sin bx B cos bx 13 The undetermined coefficients A and B can now be computed by substituting 13 into 12 and equating the resulting coef cients of sin bx and cos bx on the left and right These steps work just as well if the right side of equation 12 is replaced by cos bx or any linear combination of sin bx and cos bx that is any function of the form Ct sin bx B cos bx As before the method breaks down if 13 satis es the homogeneous equation corresponding to 12 When this happens the procedure can be carried through by using yp xA sin bx B cos bx 14 as our trial solution instead of 13 Example 1 Find a particular solution of y y sin x 15 The reduced homogeneous equation y y O has y c sinx C2 cosx as its general solution so it is useless to take y A sinx B cosx as a trial solution for the complete equation 15 We therefore try yp xA sinx B cos x This yields y39 Asinx Bcosx xA cosx Bsinx and y 2A cosx 2B sinx xA sinx Bcosx and by substituting in 15 we obtain 2A cosx 2B sinx sin x This tells us that the choice A 0 and satis es our requirement so y x cosx is the desired particular solution SECOND ORDER LINEAR EQUATIONS 103 how this works for exponentials sines and cosines and polynomials In Problem 3 we indicate a course of action for the case in which Rx is a sum of such functions It is also possible to develop slightly more elaborate techniques for handling various products of these elementary functions but for most practical purposes this is unnecessary In essence the whole matter is simply a question of intelligent guesswork involving a suf cient number of undetermined coef cients that can be tailored to t the circumstances PROBLEMS 1 Find the general solution of each of the following equations a y 3y 10y 6e by 4y 3sinx yquot 10y 25y 14equot5quot d yquot 2y 5y 25x2 12 3 Yquot r Y 6 206quot f yquot 3y 2y 14sin2x 18cos2x 8 Yquot y Zcosx h yquot 2y 123 10 i y 2y y 6e j yquot 2y 2y e sinx lit y y 10x 2 2 Ifk and b are positive constants nd the general solution of y kzy sin bx 3 If yx and y2x are solutions of yquot Pxy Qxy Rx and y Pxy Qxy R2x show that yx yx y2x is a solution of y Pxy39 Qxy Rx R2x This is called the principle of superposition Use this principle to nd the general solution of a yquot 4y 4cos2x 6cosx 8x2 4x b yquot 9y 25in 3x 4sinx 26e Z 27x3 19 THE METHOD OF VARIATION OF PARAMETERS The technique described in Section 18 for determining a particular solution of the nonhomogeneous equation y Pxy Qxy Rx 1 SECOND ORDER LINEAR EQUATIONS 105 Taking 6 and 10 together we have two equations in the two unknowns U and U5 vlyl v yl 0 viyi v y Rx These can be solved at once giving I y2Rx U1 y1Rx 1 quot 2 quot quot quot W 1 2 Wy1y2 It should be noted that these formulas are legitimate for the Wronskian in the denominators is nonzero by the linear independence of y and y2 All that remains is to integrate formulas 11 to nd v and U2 v and 11 quot 2Rx iRx 1 dx and U d 12 W 1 2 2 Wltyy2gt quot We can now put everything together and assert that yzRx y1Rx y y dx y d 13 Wy1y2 2 Wlty1y2gt quot is the particular solution of 1 we are seeking The reader will see that this method has disadvantages of its own In particular the integrals in 12 may be dif cult or impossible to work out Also of course it is necessary to know the general solution of 2 before the process can even be started but this objection is really immaterial because we are unlikely to care about nding a particular solution of 1 unless the general solution of 2 is already at hand The method of variation of parameters was invented by the French mathematician Lagrange in connection with his epochmaking work in analytical mechanics see Appendix A in Chapter 12 Example 1 Find a particular solution of y y csc x The corresponding homogeneous equation y y 0 has yx c sinx c2 cosx as its general solution so y sin x y cos x y2 cos x and y sin x The Wronskian of y and y2 is Wy1y2 my yzyi sinzx coszx 1 so by 12 we have cosx cscx cosx v fsinxdx logs1nx and U2 fsiniiscxdx x Accordingly y sinx log sin x x cosx is the desired particular solution 3 D39EFquotlquot EquotI a E l li J l 39EDUA39Tl I5395 0 Find a p r tulaar slutinn nf P U O P X 0 23 t arst by i EpxECIZi39D an tlwn tiny variatinn Dlf parameft I5 P Find a parIicuiaampr 5 lLnti0n cwf p Z V X rs by mndetarminEd r e1 icienIs and than by variartinn of u rame IerS 3 Find 3 pamicular sulutiDn Uf Each fquot the fni owimg aq ua tim5 a yquot P 4 tan P0Q d y quotquot 53 E 6quot 5 I3 yquot 239 y yM 9quot lngx E 2quot 33 y quot3quotquot J yquot E 2 O 64x quot E yquot 334 By 1 e j quot Find 3 particulajr 5D lutinn Uf each Uf the fDll 2wing neqluati sr 311 yquot y sErt ea 31quot 4 cL 2 tang pX yquot y o t3t LJ yquot 1 2 5ewt39rE n1 c yquot y mt ZD g yquot F y secs Est 151 Vd yquot y x CD5JE j 3 Shaw that the m thnd nf varatimtl Hf paIametr5 applied to the eqruatinn J C x leads tun the partiuA1ar sm1utVinn yr I 2 xf I a5in I a 7L b Find a 5im Mar fDrmua for a partil uIiar 50luwim Bf the 39EqLl Ii2l39I1 yquot U fljx where 8 is a pnitim mnsmnt g7 F inn W 39ljl1E g 1n61quota1 5r39rliutiJra of E3131 Bf the fullu wing Equati n5 3 if 9 W 5 233 2 E quotIf 11 W I3 I4F quot 4 3 39V fE 2 II ITIV W C H E I J quot 4quot IF 3 U E I39ll 1 xjgquot 1 yquot y r 331 3 y a 23 w 3939 20 VIERATID39NS AND ELECTRICAL T Genegrallly speaking 39vibrati un5 nccur when ever 3 physical systenl in 5tVafbl equi ihriuVm is di tllfrb di for thEn it is sulfnject EU IflZ3 i3j39EE5 tending Iva rmmre its aquiVlihr iium In the quotpresent E fi W6 sihall see haw situatim15 f this kinr1J can lead in differaential equati nss of fh farm dig it 0U L R E39 r P dr q I and alksi3 Ah vw quotthus smdy gf these aquat17tnV5 S39IEdS lighl an the physi nafl circu11154tan uas SECOND ORDER LINEAR EQUATIONS 107 TTYY M u 14 FIGURE 19 Undamped simple harmonic vibrations As a continuing example we consider a cart of mass M attached to a nearby wall by means of a spring Fig 19 The spring exerts no force when the cart is at its equilibrium position x 0 If the cart is displaced by a distance x then the spring exerts a restoring force F kx where k is a positive constant whose magnitude is a measure of the stiffness of the spring By Newton s second law of motion which says that the mass of the cart times its acceleration equals the total force acting on it we have dzx M dt7 F 1 0139 dzx k 3 17 0 2 It will be convenient to write this equation of motion in the form dzx i azx 0 3 112 where a VkM and its general solution can be written down at once x cl sin at C2 cos at 4 If the cart is pulled aside to the position x 2 x0 and released without any initial velocity at time t 0 so that our initial conditions are dx E0 whentO 5 x x0 and U then it is easily seen that c 2 0 and C2 x0 so 4 becomes x x0 cos at 6 The graph of 6 is shown in Fig 20 The amplitude of this simple harmonic vibration is x0 and since its period T is the time required for 108 TDquotE39FFEEE1quotFWAIEU TIC1N5 In FllGURE mite c mplete iyClE WE have 139T 23 antil W H Its fteut rcy f is the n39um br5 I Dif cycles par umitt titmc so U 1 and 8 It is dear tram B that the freqtuency stiff this vibratimnt incrca s if the stti nesst at the spring is iI1 ClI739E d at if the nfta55 of the cart is decmae dt as am mmmnn sense would have Ied US In predict t amped I Figtb IIa llm I A5 ur next step in detvteiJ0pintg this phtysicallt proh lamgi we mnsidUer the atdd39iti n I B er391quot Inf 3 ti mp E g ftzvrtr E due LG the vistDSity of tlrme mampdiutmt tthmugh whLith the cart mmvas ir wtatter 1iL etct P make ttzhe spetztt c assutmptinn thatt this fume 39Dpp t5ES the martian and has magmtudc ptf pDfti afl tn the tvetmcitty ttlmt is that ti rdxfd 39 whearte 5 is a pusitive tsatnstant f J TlE Sv11tt39it i1g tha resisttatnca mf the ttnetcitun1i Enqutati nt H nriw hEE mE E39 E9 W 11 O 39l ilt1e au39 iiE1I taqutattinn is m2 Ebm a2 v 12 SECOND ORDER LINEAR EQUATIONS 109 and its roots ml and quot12 are given by 2b 1 V4b 4 2 77117712 2 a b2ib2 a2 13 The general solution of 11 is of course determined by the nature of the numbers ml and m2 As we know there are three cases which we consider separately CASE A b2 a2 gt 0 or b gt a In loose terms this amounts to assuming that the frictional force due to the viscosity is large compared to the stiffness of the spring It follows that m and quot12 are distinct negative numbers and the general solution of 11 is x c1e39quot c2e quot2 14 If we apply the initial conditions 5 to evaluate c and C2 14 becomes x K X 7 11 31me quot2 m2e39quotquot 15 2 The graph of this function is given in Fig 21 It is clear that no vibration occurs and that the cart merely subsides to its equilibrium position This type of motion is called overdamped We now imagine that the viscosity is decreased until we reach the condition of the next case CASE B b2 a2 O or b a Here we have m1 m2 b a and the general solution of 11 is x c1e quot c2te 16 When the initial conditions 5 are imposed we obtain x x0e quot1 at 17 all FIGURE 21 III mFF EeEiTieL EQLlAquoti iD39H5 This ifusncitieim has 3 greiph siimijier tee tihet ef and again we have me H viibretiieni Any metiien f this kximzli ie seiidi lie The erii ictieiiiy iiempedi if the vieeeeity is raw cieereeeed by any amount heweivesr Smell then the metie1i3bemme5 ivihreitenri and i3 eaiieid Hmv Erd m39Ij Ed i This is the really initeresitinig siit uetien iwphieh we discuss es feiiwsi CASE 373 0 0 er b tr e Here mi and m3 are eenajugexte eemjpex numibiere 4 ie where and the genierel eeiutiein ef 11 is Jr e39 F quot39IIe39 l CD5 en C5in 13 when em and E3 ere emiiiueted in ec eerdenie e with the irnitiai ieendiitime S Iiiis ieeeewmeis xi i T V E Jf equotb eee5 at em eiri 19 1 If we intreduiee 8 E teif ibife then 194 can be expressied in I7ilE mre r39eivea5ling rrrrrn u This funetien eeeililetes with en empiitude that fails eff eizpenentially as Fig 22 eheiwsi ii is nevi peiriedie in the strict sense but its grapii ereesesi the eqiiiiibi iuim pDsitieii 1 at regule Jr i ntewei5 If we ucieinsider its iperied q as thee time reqL1ir39eid fer eniei eemipiete ey eiie2 ether eel 21 and Zn 211 225 1 bi 39 xmm i aim 213 FIG li c SECOND ORDER LINEAR EQUATIONS 111 Also its frequency f is given by f l 139c1 T 7 1 C C2 22 quotquotT 392n quotzn M 4M2 This number is usually called the natural frequency of the system When the viscosity vanishes so that c at 0 it is clear that 21 and 22 reduce to 7 and 8 Furthermore on comparing 8 and 22 we see that the frequency of the vibration is decreased by damping as we might expect Forced vibrations The vibrations discussed above are known as free vibrations because all the forces acting on the system are internal to the system itself We now extend our analysis to cover the case in which an impressed external force E f t acts on the cart Such a force might arise in many ways for example from vibrations of the wall to which the spring is attached or from the effect on the cart of an external magnetic field if the cart is made of iron In place of 9 we now have 2 dz ME EFdF 23 so 2 dx dx M Ca kx ft 24 The most important case is that in which the impressed force is periodic and has the form f t F0 cos cut so that 24 becomes dzx dx ME5c1tkxFocoswt 25 We have already solved the corresponding homogeneous equation 10 so in seeking the general solution of 25 all that remains is to nd a particular solution This is most readily accomplished by the method of undetermined coef cients Accordingly we take x A sin out B cos cut as a trial solution On substituting this into 25 we obtain the following pair of equations for A and B acA k w2MB F5 k a2MA acB 0 The solution of this system is wcF0 k w2ME A d B 2 k a2M2 a cz 3 k w2M2 wzcz Our desired particular solution is therefore F0 x k w2M2 wzcz wc sin wt k a2M cos cut 26 113 SECOND ORDER LINEAR EQUATIONS correspondences suggest themselves mass M lt gt inductance L viscosity c lt gt resistance R 1 stiffness of spring k ltgt reciprocal of capacitance E displacement x 6gt charge Q on capacitor This analogy between the mechanical and electrical systems renders identical the mathematics of the two systems and enables us to carry over at once all mathematical conclusions from the rst to the second In the given electric circuit we therefore have a critical resistance below which the free behavior of the circuit will be vibratory with a certain natural frequency a forced steady state vibration of the charge Q and resonance phenomena that appear when the circumstances are favorable PROBLEMS 1 Consider the forced vibration 27 in the underdamped case and nd the impressed frequency for which the amplitude 29 attains a maximum Will such an impressed frequency necessarily exist This value of the impressed frequency when it exists is called the resonance frequency Show that it is always less than the natural frequency Consider the underdamped free vibration described by formula 20 Show that x assumes maximum values fort 0 T 2T where Tis the period as given in formula 21 If x and x2 are any two successive maximum values of x show that xxz e T The logarithm of this quantity bT is known as the logarithmic decrement of the vibration A spherical buoy of radius r oats half submerged in water If it is depressed slightly a restoring force equal to the weight of the displaced water presses it upward and if it is then released it will bob up and down Find the period of oscillation if the friction of the water is neglected A cylindrical buoy 2 feet in diameter oats with its axis vertical in fresh water of density 624lbft3 When depressed slightly and released its period of oscillation is observed to be 19 seconds What is the weight of the buoy Suppose that a straight tunnel is drilled through the earth between any two points on the surface If tracks are laid then neglecting friction a train placed in the tunnel at one end will roll through the earth under its own weight stop at the other end and return Show that the time required for a complete round trip is the same for all such tunnels and estimate its value If the tunnel is 2L miles long what is the greatest speed attained by the train The cart in Fig 19 weighs 128 pounds and is attached to the wall by a spring with spring constant k 64 lbft The cart is pulled 6 inches in the direction away from the wall and released with no initial velocity Simultaneously a periodic external force E ft 32 sin 4t is applied to the cart Assuming that there is no air resistance nd the position x xt of the cart at time t Note particularly that xt has arbitrarily large values as t gt 00 a phenome non known as pure resonance and caused by the fact that the forcing function has the same period as the free vibrations of the unforced system 11 DItFFEjLtEi4T Ir39l E ufefft Ti Thtie ptehiem is ihtend ed tent fer stttelehte when are net iimimtiatied by eaieuletiens with tti mpleh rtt1mhere The E ttquotESp dEi 1C E hietweteh Eivqttati f 25 and 30 makes it eieefy ttiit wtit39te dewth the steadyettrete 5eitttien evf 3t he meteigr changing the tt t ti t in 2 F Eu 1 1 f f 397 quot7 39 I 77 hi pC quot 5 I Q teEL39 11332 ms EU W 1 whzere ten me teiRJ ifCi ML In eteetriiee39i etttgihtetetitittg is euetemetjt te think wet eee hit in 30 es the real part ef Enequot 39 arm tineteed hf E30 we weutid thteitft EID39I39tEiE r the di ietretttieil eqttetitJn 1 39 Iquotmet Fitttl eh pertieejier eeiutiteh ett thie eiueti e nt by the h1etheti ef undetsermtined eee ietitiehtet end at the end ef the eaieeieitient 39tait the t39E i part at this eeietieh and tIherehy ehteih the eehitieh W ef the tiiffer39ehtiei eqtietieh I3UE H The tiee ifitf E l t7lZZl4lvE3t ft U 1zbEtIl395 in the metthemet ite hef teleietriet titr etii t pmbieme wee p39itZtt It2EttEtdi his the methemetieiah it1tr39ttt i4tmt end eieetrieeI ll fr3lJgi1l lEEf Cheriee Preteee Srteinmerte tt tti5A 1923 A5 at quotDLit39lg men in Cietrmehy hie etmileht eeeieiiet eethtitiee get him inte ttreuhIe g Bi5rnhreltquot e petiee end he hastily emgrated te t5tt t1Et39iztt1 in 133939 tie quotwee vemplieteed hit the Geriehel Eleetrie E3e39mpen3r in its eetquotliielttt pertiedT et1Li he quieltiy heeeme39 the eeienti e ihreiitte ef the C 1339hn ane pre hehty the grteieteeit et ell eieetrieat e ntgiheeht Whtent he eettte te o there wee he wejF te mee5predttee eieettie rt t39 Et39EEtquot5 er gehereterei etttli he e eenntt39tttiCi alijr viehte wee te trehee1i39t etee39triie pewer mete than 3 miiiee Steinhietie 5IIl39t39IZ d these ptehteme he tteing thethemeties ehd the pewerr et c ewe t1 tt ttvd end thereby imptmed39 h39uIf1Z11 Tlife ferettet itt W3 5 tzee nuimer39et1E te ettint Itie wee e tiwett whe wee etr39ipreled his e eehgehitel defthzrmity and iiquott39t I with pein htrt he was ueieereeiiy admired fer hie eeiehti e geniue entl levee ifet hie wttrtm humertity and prttehieh S e eff httrhet The i39eilIe wihg IiEHEk39t1tjW httt uhfeitgettahle et1eedie39te ahettt hitn was pu39hI 5hee U the Lettere eeetien ef Life megltazihke hie M WEi5 5itquot ii ht jmtit arrtieie 39Itt Steirtmete f tpti yevu rtientiettet1 e ieetteui39te39tihn with iIe rtr3quot Feed rL fttthet39 Butt Sttciittt where was an eh1ptnree at Henry Ferd Ier rtittttrty greets related in me the etety hehinti that 1EE39liittg Teehnice treuhlee vde eeiepedt with e iuge new egeeereter at F7 rti395 Rieerr R llgr piehtt Hie teileetriee engineers were unable ti ieeete the dii quottt tJi39tr E513 Fete ttnIiseriltttl the aitl IJF Steittirrtete When the I ittie gient i3Tri iHquotElC at p tteht he rejteetevdi ell eeeietenieet eeitit1g etiiy fer e ne teheet peeeiit end eht Fer tee etiteightit deyei and rtigh39te he hetened te the g tt tE EvUt end tmedei eemttilveee ee4mpettatieneu Then he tt it d fer a ledtie r e meeeering tape end a pieee etquot e39heih He Nah f iEttt5ift39 eeeendeid the tiadt ier medie emeiult t39t39tEaL5tt39EITt tt1 t39t39iE end put it eheth hieIt eh the side ef the gtentereter tie elieeeehdeti and teid hie sieepitieei atitiiehee te remeve H plate them the hide hiquot the gEt lEIquot lt quotF and ill lk hut ir t1i39it ItdiIig5 item the eiel eeil at that teeetien The eerreetiehe weee t39i1 t dE39 and the genetetter then ftin li ttiiitll p tTfEvEiil39 Stehteeeeenthr time tveeeieed E hilt fer eignezd he Steinmetz ter F irti re tt1rhetI the hilt ere39hhenweegieg the geeti jeh dt39ItI TItquot quothit quot3tei1tthe39te hut I E5ZiEETfUiHfp39 reques39ting am itiemieed Sit i im f S lE39iItTt tt339LE tepiieti es fel itt t ee 139r iEi4iii l7Ig ehttiit merit eh get1etetter 1 Iithnwihg w39h tit39e ID tttttke ittttt rz quotTutti due 10 SECOND ORDER LINEAR EQUATIONS 115 21 NEWTON S LAW OF GRAVITATION AND THE MOTION OF THE PLANETS The inverse square law of attraction underlies so many natural phenomena the orbits of the planets around the sun the motion of the moon and arti cial satellites about the earth the paths described by charged particles in atomic physics etc that every person educated in science ought to know something about its consequences Our purpose in this section is to deduce Kepler s laws of planetary motion from Newton s law of universal gravitation and to this end we discuss the motion of a small particle of mass m a planet under the attraction of a xed large particle of mass M the sun For problems involving a moving particle in which the force acting on it is always directed along the line from the particle to a xed point it is usually simplest to resolve the velocity acceleration and force into components along and perpendicular to this line We therefore place the xed particle M at the origin of a polar coordinate system Fig 23 and express the radius vector from the origin to the moving particle m in the form t r ru 1 where u is the unit vector in the direction of 19 It is clear that u icos 6 jsin 6 2 r R F F9 A Fr quot1 r quot9 ur 9 M quot FIGURE 23 9 We here adopt the usual convention of signifying vectors by boldface type 116 fF IfERE bT39IAL LE UATivDN5 and E1150 that the c rrVEspm1di Vng unit vemzmr I15 perpendimlar to I11 in the dVirwac nm Bf inErEa5ing P is givcn by Ila isin 9 ms M 3 The simple relatinnAs dug id 2 I3 and nbtamed by di erexntiatmgi and 73 are assen t4iaTI fErI mmputing the velmity and acre eVratVimn U EC ifUI395 1quot and at Direct caIcL1latriun Vfmm 1 now y ie1 Lils T dr du dr i39 r f9 dr d9 d4r F Fdrgt 4 P F 3939 W r 1lg de G R m J dv E die 2dr of dlr d9 5 3 air F ms Ad dr quot5 airrl 3 TI If the farce F aCtin g an is wriquottten in the fnrmA V39 G 0 than frrmA and 0 and Nawtnm 5 second law cf Amntimn ma F we get d Edrd9 F ad Er aia A 5 P ra m ms at an 3 quot 391 dill ma 3 T4 These di femntia eqguaitiirsrns gnv rn 1hEmmi1n mf the partilE 0 andT are valid reJga1clV1e5s the nature at lzhe ifDI CE Our TE l1Equot 39 task is In ExtVram insmrA39matiun fmm Vrthem by makin1g suitablc a5umpt imns ab ut tha direciri zn and mVagn1i39tudE nf P Central fnmas and KE39pIEIf Se nmId Law F is called a cemrm fbrce it has rm c0mpnnenI perpendic u1ar In r that is if F Under rhiis as5ump1tinn the p of eq uatim1s T bmL0Vm es 9 drgr F d d r Dru mI tiplying thmulghi by r we D bt3in E E1 drd I drE t d 139r Li H 0 CIT SECOND ORDER LINEAR EQUATIONS 117 S0 d6 z dt h 8 for some constant h We shall assume that h is positive which evidently means that m is moving in a counterclockwise direction If A At is the area swept out by r from some xed position of reference so that 39 dA r2 d62 then 8 implies that 1 d6 1 dA2 2 r dt dt Zhdt 9 On integrating 9 from t to t2 we get 1 A02 quotquot A01 139 ht2 quot t1 10 This yields Kepler s second law the radius vector r from the sun to a planet sweeps out equal areas in equal intervals of time Central gravitational forces and Kepler s First Law We now specialize even further and assume that F is a central attractive force whose magnitude according to Newton s law of gravitation is directly pro portional to the product of the two masses and inversely proportional to the square of the distance between them M F Gr quot 11 The letter G represents the gravitational constant which is one of the universal constants of nature If we write 11 in the slightly simpler form km r2 F where k GM then the second of equations 7 becomes dzr d62 k 12 r dtz dt 2 The next step in this line of thought is difficult to motivate because it involves considerable technical ingenuity but we will try Our purpose I 10 When the Danish astronomer Tycho Brahe died in 1601 his assistant Johannes Kepler 15711630 inherited great masses of raw data on the positions of the planets at various times Kepler worked incessantly on this material for 20 years and at last succeeded in distilling from it his three beautifully simple laws of planetary motion which were the climax of thousands of years of purely observational astronomy DiFFEF Ei iTl L fEID39U THTIH5i is In use the diHEfB Ii3l Eql1 iiiDn 12 to Wtttttaiin th aquatic tn the orbit in ithe jp lar fnmt r 2 fm so we wa1t391t ttzi eiiminati2 E from f and cmnsider h as the iint1cpesniicntt variable Aism we want F m ht the Cl p d it vaiiriiaibiila bttt z 8 used In put 12 in the form dzr 112 39 13 tit F3 r3 quot y than HTE presence Bf p w f f Hr stuggestis that it might be tetmpmatiily ED VE iE liE to i I i1ZiquotEdltEli39 a new dteVpendEntt vatiaible z L quotr Tn aaciUmplisit titase viariimts aims wt must first t 39upIquotE5S di3rtfdt3 in terms ttf tti 3z ai6i3 by alsulat iiti dquot39 g 11F EX ii d9 139di IE at tit z iii quot zquotHt d3 zlarerl quot die an ti t P A E g d dzd6 tdilz in die at 5 i h tit id dB aft hate I I ta when trhe ilaititet Expr it is i fmd in 0V and Mr is rep Ilaced by zit we UT The g netral svztitttitm f thit EE 7iLi can be writt n drawn at DLITIEE 2 E A sin t Btcnist 8 P 14 Fm the sake of 5itmpiitcti39ty we tshit39t the dtitt ctiiun at the poviat axis in Slmh tat way tht t r is minimal quotthat is m is clisest to tha tjrigin when 6 4 TifTtilf5L39 means that 3 is 10 be imaztimait in g dircctiiittn an dz zz ten 6 U P when 9 Us THESE ct1ttdi tim1s iitmtpiy that A I D and B j If WE irmrw l39Efp13EE 2 by flirt than i1t4i can ibwze wtrittten F WA1 V WEE WhE39 B ED53 B it 1 hgfkj E05 8 SECOND ORDER LINEAR EQUATIONS 119 3 u j jj u nu j 1 T P FIGURE 24 and if we put at Bh2 k then our equation for the orbit becomes 12k lt15 r where e is a positive constant At this point we recall Fig 24 that the locus de ned by PF PD e is the conic section with focus F directrix d and eccentricity e When this condition is expressed in terms of r and 6 it is easy to see that P 1 e cos 6 is the polar equation of our conic section which is an ellipse a parabola or a hyperbola according as e lt 1 e 2 1 or e gt 1 These remarks show that the orbit 15 is a conic section with eccentricity e Bhzk and since the planets remain in the solar system and do not move in nitely far away from the sun the ellipse is the only possibility This yields Kepler s rst law the orbit of each planet is an ellipse with the sun at one focus The physical meaning of the eccentricity It follows from equation 4 that the kinetic energy of m is 1 1 d6 2 dr 2 2 quot 2 2 quotl 2 a 127 lt16 The potential energy of the system is the negative of the work required to meFeeErrrraL E 391JilAquoti39 Dir I5 rnve be Tm neity whe39re the pe1ermiel energy is mere and Z therreierr e V V H T r If X is the ietai emery ef the 53Ste rm whieii p eernetent by the prriineipfier 01 ee39meerve IiDrn ef eneergy then 16 and 1 yield P vL P E FM O P iquot3e 3 i 2 O LQ the ineiternt wherr H 0 15 and O give L L H and d T p H V I 1 e It 3L easy in efliminetner p N frem these equetiem5 and when the resuiiz is eelweui fore e we nd that ii 1 E G HE This enrebries I tie write equeitiein 15 fer three erbit in the farm jC F D em 1 P E 2h2fmk3 ens 6 A 1 It is eevidem frerrr 19 rhei the erhit is en eliipse re pereibele er a h3 quotperrbeiie eeeertdirrrg as E 1 U P er 2 p it is thergterfere eieer that the rreturre ef the eribirt ref P 7 ie EeCJfl 1j3quotiE I Ei cietrermirnred by ite tetel rergreergy E Thus the pieneite in the eeier eyeritremr hEw39 e nregetive rerrergiee and rnewer in erliipeee and bodies peeeing trhre39ugi1 the eeirer eyet39eim at htigh eprereds rhrreve gpeeitive energies and travel eiemg rhyperibeli e pethe It is ir1tere5ting lie rreeiiee ethet if e prler1et like time earth eeuid given are push frern beifrind sru ireient w etrerig te speed it up and lift its tertei energy abrerve ere weruid errter irrte e hj prE391 b39 eiiE erbit em iemwe the easier eyeetrem p ff r ti iz piereijredelt ei rievn il39u39iirrm M the fplenets and KIE E fiS Third Lam 0 new ereewiet our etteritiunr re ithe ease in wiiieh has an eiliipiie erbit 25 wheee palerquot end rereierlgelzrr eqiuetiiens are 157 and A2 J FL it ie weii krnewn frem eirementery anreiy t ie geernreetrjgr that ree the arid C1 13 bi see ea e bzjfgi end bl Mi P 20 secowo ORDER LINEAR EQUATIONS 121 NW FIGURE 25 In astronomy the semimajor axis of the orbit is called the mean distance because it is onehalf the sum of the least and greatest values of r so 15 and 20 give 1h2k hzk hz 11202 2 1ei1 e k1 e2 kbz and we have 9 k If T is the period of m that is the time required for one complete revolution in its orbit then since the area of the ellipse is Jtab it follows from 10 that Jrab hT2 In view of 21 this yields 4JI2d2b2 412 T2 3977quot TV 22 In the present idealized treatment the constant k GM depends on the central mass M but not on m so 22 holds for all the planets in our solar system and we have Keplers third law the squares of the periods of revolution of the planets are proportional to the cubes of their mean distances b2 21 The ideas of this section are of course due primarily to Newton Appendix B However the arguments given here are quite different from those that were used in print by Newton himself for he made no explicit use of the methods of calculus in any of his published works on 0 G nIssssssrriaL smJa1mHs physics or ssttrstanttirsmy Fm him cstst uIuts wss s privsts tmsthsd sf smsnttii c tintvesttigsti ts unknown to his sstntssmjpstrsrisst sndt he had tn rewrite his dtissmresisst tints the lsngttsgs sf slsstsissl gssmstryt whtstnsimr he wished ts C I39IquotzlI71Ll iiC E ts sthsrs P39R IB 1 In pra ctiua1 wstrk with Kerpl sr s third law 22 it is EUSt t 7a1quot39 tot tmsasure T in years and As in ssstronnmiss units 1 ss39tmn m39icsl unit p the sacrttfs meant distianc s w It rmtilss E 15U t ki39ltrmstsrs P these mn venimt UIi E5 sf messugr smsnt 2E takes tilts simplisr farm TE P 5amp1 What is tins psriad sf revDI u tiunt Tnff a plartezt whnss mean distance fr sm the sun is fa twtics that sf the esrth39 D stihrss t39iImss that sf the tesrtht39 2 twsnsty tvs times that sit the santht sJ Msrctutgfst yEsJr quot is dsysts 5squottLst is Msrcurfs mean distsnsee fmmt the sun 1fl The mean distame sf the pls39nst Sstumt is asstrmtnntEstst runitst Whats is SstL11rn st ps39rirmlJ sat rsssluttisn absut the sun P KEpJlfEtquot5t rst ttWD laws in the form of s ustisins 3 and 15 that 6 is s ttrs tsd t39srwsr d thus srigin with at fisrss 39whnss magni tt1 ds is isnvsrssty jpxmspiurtitmsit ts ths ssltstrts sf F Thris was Nswt un s hsndamsnt l ditscsrvenr fas it ttauSsd tn pmpur1sI391ad hiss law sf gtsvtiits lsn sud innssstigst st its muss qusnsss Prsss this tsy sstsuming E and 415 and tsmtfyitig ths fsllswitmg St tE39mEt 39E s As V til ks ms 3 is is U cc 0 F P dr ks s H i3 Esau 9 p 75 Jquotf 39 p Shttiw that the sgpsusd LI of s planets at any gp itnt sf its srtatit is gi39s39srn 12 2 1 as ts 0 tr st Suppsvss that the earth sspttrdss ints fragments which y s st tfhs same stpststi in di 39srst1t dirsquottinns ints mrbits nf their W39l I Us Ksplsrquot s 39t11ird law and that t39ssuijt sf Fmbism it to shms that all fragme1Its that do not sfs ifnts the sun sr sscspws M the sm Ear system wit rsunits ilsitsr st the sszrns p itnt where they hsgsnt to tl1itrsrg st 22 R l tt3iAE EQUsTmNs Cmustsn HAEMOVL P sc B Essen thought the main tttnpic Of this shsptsr is S nd trdssr linsst squati ns thets sfs stsvsrsl sspsiczts sf highst mdtsr limes squstimns that makes it wrstrtthgwhiuls tut ditsstuss theim btrits y SECOND ORDER LINEAR EQUATIONS 123 Most of the ideas and methods described in Sections 14 to 19 are easily extended to nth order linear equations with constant coef cients YW alyw l quot39 an ly any fx 1 where f x is assumed to be continuous on an interval ab The basic fact to keep in mind is that the general solution of 1 has the form we expect yx ygx ypx where yx is any particular solution of 1 and ygx is the general solution of the reduced homogeneous equation yo ayquot m 39 39 P a1y any n The proof is exactly the same as the proof for the case n 2 and will not be repeated e We begin by considering the problem of nding the general solution of the homogeneous equation 2 Our experience with the case n 2 tells us that this equation probably has solutions of the form y equot for suitable values of the constant r By substituting y 6 and its derivatives into 2 and dividing out the nonzero factor equot we obtain the auxiliary equation 39rquot arquotquot JT ar a O 3 The polynomial on the left side of 3 is called the auxiliary polynomial in principle it can always be factored completely into a product of n linear factors and equation 3 can then be written in the factored form I39r1rr2quotrrn0 The constants rl r2 r are the roots of the auxiliary equation 3 If these roots are distinct from one another then we have n distinct solutions equotquot e 2quot e quot 4 of the homogeneous equation 2 Just as in the case n 2 the linear combination yx cequotquot c2e ce quotquot 5 is also a solution for every choice of the coef cients c C2 c Since 5 contains n arbitrary constants we have reasonable grounds for hoping that it is the general solution of the nth order equation 2 To elevate this hope into a certainty e must appeal to a small body of theory that we now sketch very brie y When the theorems of Sections 14 and 15 are extended in the natural way it can be proved that S is the general solution of 2 if the 0N JiFFEFEEH FlAL Eu uATitm5 suiuftii0ns M are iii391E iquotl39 inidepeinde tit Thme Hire seviezrail wayst f est bi39ishirg the fact that the smiutiinnst 4 are linEar Ijr i dEpE tl39jvE T whcnuevcit th mmts r5 Ari E ii rm are tti5t39itnCt but WE Dmiiii the diEt iil5 It 39U1E FEeftJ TE fmli ws that n actuaily is ithiet genami sU39Iutim1 at 2 in thit casa Repeated mat rtt IS if the real mmsi at 3 are not all distinctj itihen tha stimlutittns 4 are lit1iEiatiiy dsetjendtaint and 55 i5 mitt tihe general S Ul39UiD t Fm EiiEa l t ipi if tr Po 3 than the part at 5 mmsistinig ttf Elem c2 e 2 beti me5 39 c2E 3quot and the twat cmti5tantst E1 and Ac becumire nine C39E5lSE I1I 1 1 T0 see what I IZIU this haipVpensa WE retail ithat in the special czasa at the Ee ili niil order equattitunii where we had tnjly itite twin mats rm and FIE we fizrnmdlt that whim FE 2 tr the S iTLlfti IZJ cleft c3equot had to ibis irieptltacad by c1e39 c2trr5i cl cErequot it can be veri ed stub5tiitu39ttitm1 that if rt T rt tir the nth Dtti t equati G n 2 than the iirtst twn terms Of 5 must be implanted by this same EHI39p39F39ESSi r Mmwa genaraitigw Z r rjg A cS rt is 3 real mat at mut litiptii ci39t W tlmti is at ktfnld r apeaiteci mutt of the auxiliary e quaiititm 0 U than quotthe mat U teetms in tihe sczzwiutiwtlrn O must be r pimecl by Tc 3 31 EVkxkJmE rI R similar I1mivl EF of S7UutiJ39 S is nteeded fillquot each imuitiplE malt nmrtii giving H Bt i1 E39BSpDtdi glje39 mdi E d farm mf tP in the next SE li n we wiii sthmw hi iw to rtibtain these E ipi E5 i Z39ln5 by teperat0Lr methndsi C mpl E3I mzmtzs Some of the wants of the auxiliairy tequatinnt H may ha mmpilegat ntiurntbets Siinice the E 2E ii EiEi IIS at 3 art reaai all mmpiet mots U39CCLEIr39 in E11 i j tgBl39iE C impleit pairs ta E and tit Eb As in the case it 2 1quotlE part of that Sfiflutgi I1 5 r0ttrespUndiitng tn tfwt such FD715 can ha iwrittitran in th MEil39139II5iiIi uquotE teal farm Equot A 05 bx B sin itt If H it and a it are t DUiEi oft multtiplicitty PI itiwnt we quotmust take etim Ag 5 tAtxi 39quot339cttsbx LEEJ H t if iJftquotI sin bx as part at the g ii tf i s iuittitm This rtqttims Ei 39 iEI1ZIifi539iquotIfi39ig tt1t SEImt 3 tnm1ncctioms 15 ef mE ammzng ii E1i3iijiSf39y39i g F12 itrmiiiai nntii39tinn5 2 iii t 1UnV i iii5hEng t1f39t hc 39Wr ntitii3 3 i39btl 5 fiTmUH and ti Lliincari inadnpEtt1tiI39t1c It sat it n iiInE tiDnS J quot139ziTItiiV 3 y39 Itj i5 Said l be Ii nmriy dquotept39nid amr if trrn mi timm can hit EI1tpTESSEd as a 39ingat c39tirnbimaititzutttu of the DihErE ainiil iinwriy indEptE2nLfE39 I it this it it Di gusting39iii In 5petti Z st thizs is tiigiitaiiyi em y in dEEid by EiIli39 j EitquotiiI Ii239Tl Equivaicntly iim39t1r dtpentdene irftltfttntifii t liat t39i 1iti vtIists tn reiatin at the farm 1 y1fx cyft39 4 vcnzy ixi 39 E in which at East anE ii that 65 is ntnt Ef3Elquot39tZlI ar1ui liraE11 inttlc39prtnd inrt means ttmt 3n y emch nilatitm irnpiies that 1 2 ms Li5I be EE39f ni SECOND oaoaa LINEAR EQUATIONS 125 Example 1 The differential equation ym 5y 4y 0 has auxiliary equation r4 5r2 4 r2 1r2 4 r 1r 1r 2r 2 0 Its general solution is therefore y cequot czequot C3492quot c4equot2 Example 2 The equation yquotquot 8y 16y O has auxiliary equation r4 8r2 16 r2 42 r 22r 22 0 so the general solution is y c c2xe2 C3 c4xequot2quotquot Example 3 The equation y 2yquot 2y 2y y O has auxiliary equation r42r32r2 2r1O or after factoring r 12r2 1 O The general solution is therefore y C c2xequot c3 cosx c sinx Example 4 Coupled hannonic oscillators Linear equations of order n gt 2 arise most often in physics by eliminating variables from simul taneous systems of second order equations We can see an example of this by linking together two simple harmonic oscillators of the kind discussed at the beginning of Section 20 Accordingly let two carts of masses m and m2 be attached to the left and right walls in Fig 26 by springs with spring constants k and k2 If there is no damping and these carts are left unconnected then when disturbed each moves with its own simple harmonic motion that is we have two independent harmonic oscillators We obtain coupled harmonic oscillators if we now connect the carts to each 2 To factor the auxiliary equation notice that r I is a root that can be found by inspection so r 1 is a factor of the auxiliary polynomial and the other factor can be found by long division 0 mtntmawtrtat quotEJLMTIDHS tirthet Thy a quot5pIquoti39tg with sptrirng mnstattt k3 as indicated in the tgutfet apip1ting Newttrtt S EiECDIJ39Id law naf tl t1CI tiUITl it can be sthmvn Pmb temt 136 that that diSp l E2E 1E 39tf5 12 and x of the carts atatttistty the fmttlowiing Sii 1t1l t nE I1 sfytstem nf 5Emntd 39 rdiET linear e139uattiintn5 EX 391 aft 3913 k lIt ka xa 39 j F q NI 2 dt 5 k339It if klIE kgquotti quotquotquot k3Il We ti quot39W ttb39tai11 a tsingle tottder vCq11H tiDH far 1 by 5t2tlquott39irtg that first 3q39uatit1I vn for 3 and Eub5titutirtgt in the secnd E lltl lfj P r0bItem t We hmre tl yet adr1IESEEE1 the prttb em Hf nding at particular StE3t39lItiE3I1 far the complete Equ ti In this Ct II39tt t t it stuf ces to I m i tthat the tttetlmdt Df und tietmtinedt E ffllx i is EH4S39E l1553d in S lziti t 18 CtJnt39inL1 5 ta apply twith 7bvViEtut5 t139l 7ItiIZlE3l quot hantgtcst far functimts f 5 f the L G ll lidEtEtjt int that 5ectiatJtt 113 BZ Et tsecrti nt we Shtallt examine a ttnttaltyt di etentt t pf ith ta the pmizrliem of ttding Vp39 39t39t1iEt11i l S lutti n Eamplve 53 Find 3 pattivtuIatr 5 lutimnt naf the d1i et39 3nttial EqLtHIi t y quot y tr 3 v 11 ur eitpEtiEt1tie in S Ecti0n 13 s39ugg 5t5 that quotHEB takcz 31 ttit1li E luti n of the EFITIA F xftt It a1tt Ix39l I E aux tax I quot1X393 TH Sit1tte y n an A2319 3 39gI39 yquot Eat nix 139Jd 2 639371 tautbe StitutiDH in the giwtt Evqu3tti nrt yiteldss EH 2392r1 6u2x mg Eayxt 3 3I2 313 t o IV 1rt afttI cmltccting m icicntts tf like pttwexrsz of x 39339 jxE tt2al 12 a 4amp1 A EH3 Ex E X 1 Thuat II I Lm E33 4n 6amp3 1 50 H1 7 at E c and ct 239r39 We t WT1 lquotE ftI39 II Ei E13 JE at par39tiEt1attt SlZtt 11 Ei I39I 39 0Z E27 5x1 EH33 127 SECOND ORDER LINEAR EQUATIONS PROBLEMS Find the general solution of each of the following equations 1 I5b m35F3Pws9w99w 17 18 19 20 wwow2yn y quot 3y 4y 2y 0 y y0 ymyn y VyQ y quot 4yquot 6yquot 4y y 0 y y 0 y 5y 4y 0 y Zazy a y O y Zazy aquoty 0 y 2yquot 2y 2y y 0 y 2y 2yquot 6y 5y 0 yquot 6y 11y 6y 0 y quot y 3y Sy 2y 0 y 5 6y quot 8yquot 48y 16y 96y 0 Derive equations 6 for the coupled harmonic oscillators by using the con guration shown in Fig 26 where both carts are displaced to the right from their equilibrium positions and x2 gt xi so that the spring on the right is compressed and the other two are stretched In Example 4 nd the fourth order differential equation for x by eliminating x2 as suggested In the preceding problem solve the fourth order equation for x if the masses are equal and the spring constants are equal so that m m2 m and k k2 r k3 k In this special case show directly that is without using the symmetry of the situation that x2 satis es the same differential equation as x The two frequencies associated with these coupled harmonic oscillators are called the normal frequencies of the system What are they Find the general solution ofym 0 Ofy quot sinx 24 Find the general solution ofyquot39 3y 2y 10 42423 lt In FIGURE 26 Z DiFFtE39REHTMLEDUAquot 1quotigtDNS 21 Find the 5ehitien ef yquotquot H y 1 that 5e tie39 ee the irnitiei eenditier1e y05I yw NH 4 Shew that the eherige ef 39imIejpendten39ti iveriehite 1 equot transferm5 the third ereueri Euler equmireerieienel eqL1at39ie n 13y39 39 emr2y39 e392i1yquot egy El irite e thir erder linear tequetiee with eer1etant eereEtcierrte This transfer metien else 39werhe fer the with etrtiier Enter eq ua Liien Selve the fI lewir1ge eque tieres by this reethecl S r3yquot 3x2yquot 390 0 E x39P quot Ex2yquot xy H y S bl re are 2 or 23 In tietermining the drag en a small sphere mteving at a eeneteet speed thr39eugh e vieetees 39 tlidi it is neeretssery te sewer the ei erential equetien I3Fi 8xlFrr L P O H eK Nvetieie that this is en Euler req uetier1 fer y and use the methed ef Pirehlrern 22 ten shew that the getrlerei eeietiee is 3quot eerie alxquot te3e 3 11 e4 Theee ideas are part ef the methemetieI F1eritgreuriril ueetl by Rehert A Milliikan in hie fermeue ei391drterp experiment ef l93quot9 fer meeeuring the charge en en eleetrein fer wh ieh he wen the 1923 Nehet Prize 23 0PiERATDR METHODS FOR FlNDtIifG PARTICUALALR SOLUTIONS Art the end ef Seetien 22 we referred ter the problem ref nding pertiteuier se letiens fer U39HiU39 1UgE EU39US equetietne ef the ferrn P 5 In 6 dx H In 39this eeetiein we give a very brief sketch ef the uee ef vii39iiii fE ii i eperatetr5 fer self ing this emblem in mere e icient ways then any we have seen eefere These eperatieniet 1i1ethede are niainiy ue tie the E ngIieh eppli etd metheiiietieiem Diiver t eevieviide fl 85tD 39i 925 Heeivisidtaeie imethede eeerned ee strange te the ereientiete ef his time that he wee widely regarded as 3 hereckpenitg whieh eunfer tuneteliy is re eernmee fate efer ithiehers ef iunueual erigieeilityt air 1 8 quot fH1 4 Elna ftiffl 1 13 Fer e eleer epler1etien et this exeeeeihgtyi ingeruien15 l 39 Ii3II3i39iI39I JilE3LquotlI w iIh e geed cir39aw1insg ef the eppaerettus see ip E ia i ef the heek by Lines F39eeting mirentieinercl in Se ctien 4 Nete 12 SECOND ORDER LINEAR EQUATIONS 129 Let us represent derivatives by powers of D so that dy dzy dquoty D 2 D2 9 quot Y dx y dxz D y dxn Then 1 can be written as Dquoty aDquotquoty aDy any fx 2 or as Dquot a1Dquot a1D ay fx or as pDy f x 3 where the differential operator pD is simply the auxiliary polynomial pr with r replaced by D The successive application of two or more such operators can be made by first multiplying the operators together by the usual rules of algebra and then applying the product operator For example we know that pD can be formally factored into PD D quot1D quot V2 39 39 D E1 4 where r1 r2 r are the roots of the auxiliary equation and these factors can then be applied successively in any order to yield the same result as a single application of pD As an illustration of this idea we point out that if the auxiliary equation is of the second decree and therefore has only two roots r and r2 then formally we have D quot 1D V2 2 D2 quot V1 quot2D l quotN2 5 and since d d D 72 quot 72 39 39 7392 we can verify 5 by writing D V1D 392 Z quot1 72 d a d 72 r1 X quot 72 dx dx dx dzy dy 3 391 f g 1V2 D2 r1 r2Dy rlrzy D2 quot F1 I 2D I 1I 2y for this is the meaning of 5 We have no difficulty with the meaning of the expression pDy on the left of 3 it has the same meaning as the left side of 2 or 1 Our 130 39DIFFEREN39I39I aTL EQUA39I39I iNS purpeset new is te leem hew tee treat pf as a separate entttitgg and in doing this tee devetep the emtetthed s for sewing 3 their are the eubjeet ef this section Witheut beating ereund the bush we witeh te solve fermellyquot fet1r in 3 ebteining 1 J I 5 Here 139pquotD repreeet1te an eperatien tn be perfemmedt mt pq ma yitetdt y 0 que3 ttien is what is the netutre Inf this EtEE1Ii39D39l and hnew earl We carry it out In etrder te begin te understand these matters we 39CUIlSidtBfI39 the simple equtetient D l whieh gives 1 A Jr Eft But 3 f xf er equi39velt er1tl tdyfdx 2 f r is eeeily SDIVBd by writing pm so it ie I1 Il1f l tn metnie the tdte niti1en Erefewe U This tells us that the epkereter 1 D applied te e fU1I1CIi1U means irltegrate the futnetiten Similattrly the eperetm MD applied ter a futntetien meane integrate the f39t1 Ii1I3939tii 2111 39twieee in sueeeeeien ems se en Uperatm e like MD and 111 D2 t1139e eelled inveese epere Ierr5a We eentiynuie thie lime ef irwestiga then and eamiI391e ether irwereet ep eratere Cetnetitier DrWfL e wheire r is H eenetent Feirmaltlyt we have Me T D But is the Simple ret nrder Eineer E L1 tiDI1 ry fJ wherse e 0lutient by Seetitm 17 is e e air We suppress eernetente wef integmttien bz i ul we agree enly eeelting SECOND ORDER LINEAR EQUATIONS 131 particular solutions It is therefore natural to make the de nition 1 D r fUe e 7Uhh a Notice that this reduces to 7 when r 0 We are now ready to begin carrying out the problemsolving procedures that arise from 6 METHOD 1 SUCCESSIVE INTEGRATIONS By using the factorization 4 we can write formula 6 as 1 1 mm awwmDmwDagt 1 1 1 fx fx C O I D r1D r2 Dr Here we may apply the n inverse operators in any convenient order and by 9 we know that the complete process requires n successive integrations That the resulting function y yx is a particular solution of 3 is easily seen for by applying to y the factors of pD in suitable order we undo the successive integrations and arrive back at f x Example 1 Find a particular solution of yquot 3y 2y xe Solution We have D2 3D 2y xe so 1 1 D 1D2yxe and yD1D2xequot By 9 and an integration by parts we obtain 1 s D zxe e Je39 xequotdx 1 xequot so y Iii1 xequot e fequot 1 xe dx 1 x2equot 1 Example 2 Find a particular solution of yquot y e Solution We have D2 1y 42quot so nnFEnEHTnm F UATlfH5 w PzL p The suce5sive intregmtimms of mmhcvd 11 arr likEl3quot ta bemmg cump4l4icatE ancll Iim cnVnsuming I carryr out The fn rr11rula quot X s t an E S r sugggests 3 way In amild this 39wunrk fm it suggesrlis the possibility nf decDm ptJ3ing the npBm39amr an the right inm pmrtia J fractiumas If the factasaltrs mt pD are distincL we can Awrite 1 t a AMWAI Tjfilel S W i J pym D D E p 9 rf far suimfble mnstants A11 A2 E A and each term an mh right can be f ll di by Aursin J VpB139amr in brackt5 here is 5Umet imes ca cdi the H E 1Ji dE expu n3 mV af the inm rse npEra t r Np pV Ezmmme Salve Ih 1E39 pmhIrm in E39 amp39le 1 by mmghud St i tfi M We hmre p 1 y 3quot DH1ID 2 3 2 D 1J 1 1 D E 211 D K ilE ll JE lIEI dquotI Elm rig Milk 139g 39 w xvEquot39 E EU1 1 The SIL1E1EI1t wi l nrJti E than rtlhi2 aJ1LmtiJn i5 nut quigte the same as 115 surlutitrgnjn fuund in Example B Huiweve39r it PB 3353quot In sass that quotthey d er r 1r39 by an 5DluVtiicm uf thzg rveued hnmngenems EqiL1 39EisU am an is well quotEsmmpie 439 S lwa Eh pmbhm in A gmmple 5 by this methd S39mTu azn We haeve 1 I P 9 P 1 1 A CV we 3 J a D 9 1 D Lmp 1 y Egg Je e39 dfx a Je aquot39dx Os I l 2 le gxe If sstmme Hf the facmrs of q6 are rkepeattad then we kimw thaw mm fitLrniil DEF th pa r39Ii1 fmm39E ns dECE pISili n is di Ef393IV Fm e ein1plc if w E r is El kf ld rptfa tcAci famVcJr than the dcmmp 5iti11n c0nta im5 the SECOND ORDER LINEAR EQUATIONS 133 terms A1 A2 Ak e pm D quot 71 D quot12 D 7391k These operators can be applied to f x in order from left to right each requiring an integration based on the result of the preceding step as in method 1 METHOD 3 SERIES EXPANSIONS OF OPERATORS For problems in which f x is a polynomial it is often useful to expand the inverse operator 1pD in a power series in D so that y fx 1 b1D 12202 fx pD The reason for this is that high derivatives of polynomials disappear because Dkx 0 if k gt n Example 5 Find a particular solution of y 2yquot y x 2x 5 Solution We have D3 2D2 1y x 2x 5 so 1 D3xquot2x 5 gt i 39339DTquot By ordinary long division we nd that 1 12D2D34D4 4D539 S0 y12D2 D34D4 4D5x42x5 x 2x 5 212x2 24x 424 x424x2 22x 101 In order to make the fullest use of this method it is desirable to keep in mind the following series expansions from elementary algebra 1 1 1rr2r3 and 1 r 1r 1 rr2 r3 In this context we are only interested in these formulas as formal series expansions and have no need to concern ourselves with their conver gence behavior Example 6 Find a particular solution of yquot y y y x5 2x2 36 y4 tEt39FFEHEttI I tA1E ifJUA7t1t t it5 Sat39tuH39van WE that2 D3 1 5 E At so 2139 1 D xi 0 F B t 1t D I D D 2 4 r SE at 11 5 E It 115 E 13 x 93 43 1 2 5 16 1 I l 239 mm 5 E at 4 3 125 1215 Tim F39Ei ZT 7 b lE abrm39t the prtmadiures i MS39T IEij 0f7 these xtamtptles is thattt that atctuallyt wmrkt 45 w4 pi8 7 As we 0 ampxtpn nctmt tial ftunctti mt5 bEha ti39E in at Sii 39El392fi m way unttdrer dti ErtEnti UI1t 6 fttatctt Ettaquotbies ULE tn sitmtplify tZt39tJ1 wtml t whetnevier fr mntait1tt a ftanzrtcrrt If the fmm 3quot Thus if 1 g equot tgx4t wa begin by nnticing that D G e Dgtx t39ie t39quot gxV K tre gtrI E f it t r39gt F2 p I 39 Vim WE SEE that fD1 the p ty Uf i l EtpEI39atlquot e tptfJttH gtttxt quot pDt kZIgLrt 10 This says that we can tmmte tthe factttnr 9 tea the lefttt nzf the 39fIpEF t39E33939LF prf jr if we rerp ItatEe m D in time mznpetatnrt T ht samte pf ip tftijt is valid for invetr5tet npertzatm 1fp J By apptyitntg this fnrmtuta tum tthe tsutccessive tfsactms 1 E tr D Y 2 fix an J M Tu see tthtis WE Simply pt tn the rightt side and uSE 1tJ pm EII V pEm Matti 2 M s Pmpert ics Ill and II are called hit exputnentiaf rufet Thtyr are tuscftult in mcwintg tatttnonentttital frutncttims amt 0f the Way of ntplerattnrstt SECOND ORDER LINEAR EQUATIONS w Example 7 Solve the problem in Example 1 by this method Solution We have D2 3D 2y xe so 1 X X 1 quotE77i7FequoteD1f xD 2 6 1 I1 1 x e I D2D D1Dx X1 2 e I 1DD x e x2 x 1 as we have already seen in Examples 1 and 3 Interested readers will nd additional material on the methods of this section in the Historical Introduction to H S Carslaw and J C Jaeger Operational Methods In Applied Mathematics Dover New York 1963 and in E Stephens The Elementary Theory of Operational Mathematics McGrawHill New York 1937 PROBLEMS 1 Find a particular solution ofy 4y ez by using each of Methods 1 and 2 2 Find a particular solution of y y xzez by using each of Methods 1 2 and 4 In Problems 3 to 6 nd a particular solution by using Method 1 3 y 4y 4y 10x e39Zquot 4 y 2y y e 5 y y equot 6 yquot 2y 3y 665 In Problems 7 to 15 nd a particular solution by using Method 3 y y39yx3 3x21 yquotquot 2y yZx3 3x24x5 4yquot y x4 10 y y 39 x2 11 y y x1 12 y y y3xx4 13 yquot y 1 x4 14 y 39 y 12x 2 15 y y 9x2 2x 1 99 quot 136 DIFFEREH I IAL imLaTiaHa In Pmilama La 13 r sml a hpEZrtiE39i1i aaliuitiimn by uaiing Meithad 116 quotquot 4 0 Irja1quoti IT 3quot Q a1 ia339 39 ail y39 39 By quot13 3a1 iii In Pirublama i9 ta 24 nd a pairtiic1iiaVr s iutfim39f1 by any 39ma39nhuidi 19 3939quot E E 1 ii61a2i ll yw E 1 xii p 7 p 22 3 g aquot PZE yquotquot ii a h g h y quot 2yquot 2 P USE the waapicancntiia1 shift ruia ta nd this genera aniutjicrni af aaah cf th fnli wi mg eiquatimia aft E E ihim muItiipi39j by 3quot and mac iliA D iquot2a quot quot a s P 2 E1 ainx P C 0naiiai the rail ardar hramuganaoiiaa EqLi tiiDi1hp D 1r 7 tlfafl if a pai1ma miai qr is a faziaar Df the aiuiailiary poliynamiai pfr sham that any si liiitmni of ma idiii 7ariamiai azqiuiatinin q y D ia aiau a a iutimn Bf PiDii39 0 h if r is a irnxat mi miuitiipiiiiaiiity k of ma al1N iiTi rj aquatim pm IEII ahnw that ainy EUiui39iiJn at D E rigy 5 U alan a aulutimi Iaif p EJfy ah Lisa the Eap0rmeritial shift r u1r tcn ahaw that D ryjky E D haa y CH 2at 2 cgxi W i a quoti as its general suIutiumi H39im r D P Equiwa1am in Er iz Hi E rmlty u Lauinhard Euler 1U ii39v Ei3i wag Swiitaariianifs foiriagmoat aciieniiai and nine E1ff the three greatest miathamatiaiaina of mmiarm tiimea the other twn baing Gauaa and Riama1mi HE W33 fp fh ips tha moat PfC7J iii C author f ali tirme in any i aicii From 172 in N83 his writings p uradi nut in a aaiaminigiy endless i imiida constantly aidfding knUwiadga to every kgmwn ilaranch of puira and applied maIhamiaitica amii aim in many that were mat ianawn Iumiiil has created them He averaged aibua ui print amp ii piagaai a year tihimugh Dui39I his lung iifa and yet he aimaai aiwaya haidi S fm iiili g wmihiwhiiiia E1053 ianid never aaains lacngiwindJad 39ii39ih 3 pubiica ii0in f fhia mmplaia WEJquotiiC5 wias srtartedi in 1911 and tha and is rmi in sight u5 aditin was piam1ad ID incIuda 87 titles iii 4 quotn 39DiilI39I39lES birut since that tinaa aaiianaiva new daigmsita Uf prxaviimuaiijg Lmkrmwn ITi LnL1SCI ip IEi i1aw39a lrjsaan unaartihada anti it is mrzw srscowo ORDER LINEAR EQUATIONS 137 estimated that more than 100 large volumes will be required for completion of the project Euler evidently wrote mathematics with the ease and uency of a skilled speaker discoursing on subjects with which he is intimately familiar His writings are models of relaxed clarity He never condensed and he reveled in the rich abundance of his ideas and the vast scope of his interests The French physicist Arago in speaking of Euler s incomparable mathematical facility remarked that He calcu lated without apparent effort as men breathe or as eagles sustain themselves in the wind He suffered total blindness during the last 17 years of his life but with the aid of his powerful memory and fertile imagination and with helpers to write his books and scienti c papers from dictation he actually increased his already prodigious output of work Euler was a native of Basel and a student of John Bernoulli at the University but he soon outstripped his teacher His working life was spent as a member of the Academies of Science at Berlin and St Petersburg and most of his papers were published in the journals of these organizations His business was mathematical research and he knew his business He was also a man of broad culture well versed in the classical languages and literatures he knew the Aeneid by heart many modern languages physiology medicine botany geography and the entire body of physical science as it was known in his time However he had little talent for metaphysics or disputation and came out second best in many good natured verbal encounters with Voltaire at the court of Frederick the Great His personal life was as placid and uneventful as is possible for a man with 13 children Though he was not himself a teacher Euler has had a deeper in uence on the teaching of mathematics than any other man This came about chie y through his three great treatises Introductio in Analysin In nitorum 1748 Institutiones Calculi Differentialis 1755 and Institutiones Calculi Integralis 17681794 There is considerable truth in the old saying that all elementary and advanced calculus textbooks since 1748 are essentially copies of Euler or copies of copies of Euler These works summed up and codi ed the discoveries of his predecessors and are full of Euler s own ideas He extended and perfected plane and solid analytic geometry introduced the analytic approach to trigonometry and was responsible for the modern treatment of the functions logx and ex He created a consistent theory of logarithms of negative and imaginary numbers and discovered that logx has an in nite number of values It 394 See C B Boyer The Foremost Textbook of Modern Times Am Math Monthly Vol 58 pp 223226 1951 13 t 1s39ssteEH39TttsL EmusTtrmrus was threugh his teeth that the s j tmbuts es IE and ti 2 heesme eerum en eu39rreuey fer sit methems t ieitsinsj tenet it was he whee li k39Bd them tutgether in the ssteuisthtitug retstieut em 1 This is merelyquot e special case putt 51 st his fernsus turmruls em cuts c E shim u wihtiehs entneets the tespu t1etntis 1 and trigeuemtetri e funetieutss end is shsetuete339 iudispenseb e in higher teus ty stisquot5 AmDug his ether eeutributiuns ts stsnderd smetthemsttesill uutetitutns were sin 3 c eess the use at f x fer suit usnspeet ed ttuutetmtn strut the use of fer s tummstiuu39 539 Creed mtttatjitjltt are impurtsmts hut the tidees behind them are whet restty euuut sud in this respect Euler Ls ifertiftisty was slruest heyend beliefs He pteferredt cenerete sp ecis Apmblems ID the general thteurtets in tsegue musty and unique itnstsguhtt tutu the ceunteetieus hetweetnt s pparEgttt jt uturnetsted fer mules httssed mum trscils into new arses uf mst hemstirst whiten he left fer heis stueeessers the eutttisst39es Hie west the rst end grestesltt master ef tiu uite series in nite gpreduetts sued eentinued fraetsieus and c weeks are ersmmed uriths striking d5ViSC JEtiESv in tlhese t eulds James Beruuullli Je htuquot s 39 tltleff hrutuher teuuud the sums sf S ttEtquot l in nite seriest hut he west net ehte tau t udi the sum sf the trseeipsumlst ef the squares 1 7 p IIe wrute If sumeene tsheutld succeed in nding this sum and will tell me about I shall he mush ehutied te i In L leng efter Jemes s death Euiieir meets the weudertfuhl distteetvesry that O 1 sea 1 ss1s em 1 He else found the sums sf treeipreesls of the fuurth tend sisths DWBES at H as and lquotrJ 5quot 39539 1393quot 53 D I w quotIE aha LI 15 An Equotquot llTtt39l mere sstunishh1g eeusequet1e e eif his fermuls is the feet that en imsgiusty pewter ef en imeginstjs numtue39r can be i39esl in ps39rtts39Ltist I s quot t er if we put H sf2 we ehtein e F se E tEJI nfEFJi 1Etu393nquot T E mt Eulier further 5w11Il ii39E that E hes riu ntitetljr msnyr mines emf whirh ttus eeieutetien prrutjluees sent she H See Csjuti 0 z Hfttur39 sf Mufheututafeut Nhet ffues Open 39C39u39urtg Chteegee 1929 SECOND ORDER LINEAR EQUATIONS 139 When John heard about these feats he wrote If only my brother were alive now 7 Few would believe that these formulas are related as they are to Wallis s in nite product 1656 Jr 2 E f 4 6 6 2quot3913933557 Euler was the rst to explain this in a satisfactory way in terms of his in nite product expansion of the sine 2 t2 2 S11 1 i1321 x It 4n 9 Wallis s product is also related to Brouncker s remarkable continued fraction o JI 1 12 6 32 52 72 1 2 2 2 which became understandable only in the context of Euler s extensive researches in this eld His work in all departments of analysis strongly in uenced the further development of this subject through the next two centuries He contributed many important ideas to differential equations including substantial parts of the theory of second order linear equations and the method of solution by power series He gave the rst systematic discussion of the calculus of variations which he founded on his basic differential equation for a minimizing curve He introduced the number now known as Euler s constant 1 1 1 ylim1 logn05772 2 3 n n oo which is the most important special number in mathematics after 1 and e He discovered the integral de ning the gamma function 39 Fx f tquotquot1equot dt 0 quotThe world is still waiting more than 200 years later for someone to discover the sum of the reciprocals of the cubes 140 Dtt TF EEtEtrr39LeLt ererserresrs whiesh is eiten the rst ei the ser eailser d higher ttrtahrs eehd enitei fquotIilI39EtiICiitS students meet heyhnd the iesrel ef eeieuluts and he devreieped merry ef its apiiestierrs and speetiel reperties He allse weriked with Feurierr seriesi enxeeusnteretti Bessel furretiehs in his srtiudjr ef the vib retiehs ef as stretehed circular rrremhrane end ejppiiedt Laplace trvantsfrms rte sehre dli ieriehtiei equetieris eli befere FD M1iE39I EesseL and Leplceree were b39lquoti39i iErrer1 tfheugh iEuiler died ebeut T yeeirs egve he lives everywhterier in er1sfljrsis Ti Bell the weiknse39wh rhistrrrian hf rnathemetiesi DiDSE I VEd that rte ef the meet rtemterkebie features ref Euier s universal gerquotnitus was eq39uall strentgth in hethi of the meirt eurirehts ef methemetises the EDi tiI39i39t1Cti1S and the disseriretei In the reellm erf the ciiserete he was he ef the erigineters est mederr1 humher theery end rrnerde merry fer rtesehirr eerttriihutiehs tee 0 subjveet threhgheut P8 life In edditiernt the eriihs ef tepeieg5ehre ei the dmriihant ferees ih rrrederrn rnsthsemtstiamsiiiie in his seltrtien ef the Kffurnieigsberg hridge prehiem end his fermute Z E F ti 39 nE ii g the hunters ef rertiees evdgesi end feees ef er e pelyherirerrr In the fellewihg ptsr egrta hsrj we brie y deserie serhe ref his eetI39wi ti es in these elds In rmmhetr th eer3g Euler drewi rrmeh ei irrspirstiern item the Ci i iIEgiIig marginal rtetes ieft by Ferrhst in his eepyi ef the werrksr ext Dieprherrtustt He gave the rst published preeis ef heth iFrerrmet39st theerem anti Fetrrr1et s ttwe squsrres tfheeiremt He ieter genseraiisetdi7 the first eti these relessie retsxuitsr irrtrcliueisng the Euler he hrhetiern his preef ef the seeend CrtZ ISiL him 3937 years of iriterrrnitteurrt e ierrit In edditien he pressed that every pesiitieet integer is e sum of feur squares eind iinrestiget edT iew ef quadratic i ECip1quot EiIjf ef his riniti er esttirrg werik was eetthnreeiteei with the 5EqiiEfii 2Egt ef prim mrsirriiibersi that isr with these integers 0 1 whese Ui I it positive dhivxistersr ere 1 and use et the dissergenee ef the hernrerrie series I P te prere Eueiidisr theererrr that there are 39it39tfiini iEiquot merrjyr primes is she simfle and ingiti39t39quotliUL1e that we vtehture he give it here Suppesre that there are reiniiy prhries say rrV pg P Then each integer n 3 1 uniquelsr espressihie its the ferrrl rt p per If e is the iergesrt ef these respenents these it is easy the see that i 1 4 t P in r A 1 1 39 r 39 quot quoti P 39 r 39 39 1 r quot 5 t pm M M hy m ui tipi1ri ntg ml the feeters en the r39ighti But the sim39pie i39e rmhie I 5 JE2 r ifrquot etjljlj whieh is sei ittl fer iri it 1 shews that the SECOND ORDER LINEAR EQUATIONS 141 factors in the above product are less than the numbers 1 1 1 1 1p1 11p2 quot1 1pN so 1 1 1 1lt P1 P2 PM 2 3 n p1 1p21 pN 1 for every n This contradicts the divergence of the harmonic series and shows that there cannot exist only a nite number of primes He also proved that the series 2 3 5 7 11 13 17 of the reciprocals of the primes diverges and discovered the following wonderful identity if s gt 1 then 1 3 n1 11 1 P 1 quot 1175 where the expression on the right denotes the product of the numbers 1 175 for all primes p We shall return to this identity later in our note on Riemann in Appendix E in Chapter 5 He also initiated the theory of partitions a little known branch of number theory that turned out much later to have applications in statistical mechanics and the kinetic theory of gases A typical problem of this subject is to determine the number pn of ways in which a given positive integer n can be expressed as a sum of positive integers and if possible to discover some properties of this function For example 4 can be partitioned into 431222111111 so p4 5 and similarly p539 1 7 and p6 11 It is clear that pn increases very rapidly with n so rapidly in fact that p200 39729990293ss Euler began his investigations by noticing only geniuses notice such things that pn is the coefficient of xquot when the function This evaluation required a month39s work by a skilled computer in 1918 His motive was to check an approximate formula for pn namely err 2n3 pm E 4n1T the error was extremely small 142 tiFFae EHT391a L EGLJATMDHS ET The Ksiitlig b tg leiridgteei 1 tIZji E itEl E T1 ie tetxpattded in at pewter jseriee 1 t1 HrWtH By iaetilditng an this feundattiem he der39ivera mam either rvematiktabie idetntitiea nelatede ate a W quotrWEVEF re ii prebiett1a abcmtt ptair ttitieeneiquotquot The Kt nigsbetg btidete pl Di3I39iEmt eriegitnat ed aa a pa539tirtte ef Srungdaty etreilera in the tewrt ef Kenitgaiberg tte w Kaiimngrtad in what wast ifetmeetiiy East Pruesiaeet There we re SEv E ibriievgtea aetreae the riv er that iiews ti1rettgh the tewn aete 27 The residents eased te enjey wtaiking iit39Dtit etie laattit ta the igaiainde anti tttett te the ethet banit and baelt tagaiitn a the eenviietien was iwidtetiy held that is impeatsibire tea do this eraeaing aii SE1 39et391 bridges Wi39Fii DtJ ereeaing tam bridge iTIiU Iquot than tquottCE E uiet iEi 3tifTZE39 i the prebilem by etamitting the eehematite EiifI3gt7 i39IZquotI givett an the tquotiigi lii in the gure in whieh the Qiand areas are tepreaentedt tn339 p i quott39iS and the meideigteet iinea mnntetetineg these pquotetint5t The pDiIITt391ISt are called vertieee and a uquot 139E3it is satid D be ttdd or even aeeetrdiIfIg as ti 1 e mtmiirer ef lines ieatiitnig tea it is add are Ev tt In mtetlern terminei1egyz the entire etaniiguratiten is eaified ta grepitj and El path threugeh the graph that ftraveireee every Iitfte but an iinre mere than venteew ie etailetl an E uier patht Ant Euler pa Eh need mitt end at the uquotEfT lZEnt whete it began but if it detest it is eaii ed an Ettie39r eiretaitl By the tztee vef etJntbieittatDttiaii teaaat1ingi Euiet39 arrive at the fetitieiwimg theereme abettt any Sttchi grah ii there are an E1mmxma pea Ma W ee Ch 39pquotLefi XIX ef e H i iatr dt anti Wirightt Ant iatreetut39ient te the Theerj39 elf Nttm bert iiiiJF Eii Liniverai3 t3t Pr39eaa quotE9Z39iEt er CiIaptttT5 12 l I tit c Aeertetwei i eutmtbe39r 39Th39eetry W V f auttdierag LSan Franeieeer 19quotiquoti Theae treatrmtttta are quot eiementattaquot39 in the teehnrieai 5enee tha39t they edit net use the hiighmptiwered I i1 Eh39i539JE39jf Iii 39llir i1EEKJ a alfgaiae butt E iE 1Ef ii39i39ELit2SS 39i t39IE jF ilttlv iat jituzm it t139tie Liquotquoter39 ettit1et139te whe wiah te experience aemet atquot i EuIerquota meet iFtE39aftE 5itit tg teeth in t ttH39l39i39JIr theery at ret itlaindt anti in a eInteitt t39t 39it t39tq39lIiiIquoti l391Ig mtitirh ptervi eL1ia knewIedgei we T tllm ii xdt Chapter W at G Pet3ta a nI39ttquot beeic Tedeettien end Arteiegy in tMxfifhEm IiL7j i Piquotif1tE39iUAi39i Unittet aity Press 1954 SECOND ORDER LINEAR EQUATIONS X even number of odd vertices 2 if there are no odd vertices there is an Euler circuit starting at any point 3 if there are two odd vertices there is no Euler circuit but there is an Euler path starting at one odd vertex and ending at the other 4 if there are more than two odd vertices there are no Euler paths The graph of the Konigsberg bridges has four odd vertices and therefore by the last theorem has no Euler paths The branch of mathematics that has developed from these ideas is known as graph theory it has applications to chemical bonding economics psychosociology the properties of networks of roads and railroads and other subjects A polyhedron is a solid whose surface consists of a number of polygonal faces and a regular polyhedron has faces that are regular polygons As we know there exists a regular polygon with n sides for each positive integer n 3 4 5 and they even have special names equilateral triangle square regular pentagon etc However it is a curious fact and has been known since the time of the ancient Greeks that there are only ve regular polyhedra those shown in Fig 28 with names given in the table below The Greeks studied these gures assiduously but it remained for 7 4 FIGURE 28 Regular polyhedra 20 Euler s original paper of 1736 is interesting to read and easy to understand it can be found on pp 573580 of J R Newman ed The World of Mathematics Simon and Schuster New York 1956 2 It is easy to see without appealing to any theorems that this graph contains noEuler circuit for if there were such a circuit it would have to enter each vertex as many times as it leaves it and therefore every vertex would have to be even Similar reasoning shows also that if there were an Euler path that is not a circuit there would be two odd vertices p X DIFFERENTIAL E 0u AVTu H5 Euiler ta d i5 cnvver the siimgjpi est f thcir mmmm39a prnpertiVes If V E X F IEIEITIUIB the Vnumhers of vEVrtices Edge3 and faces of any ne of N than in 6wary case we h avE This fact is knmin as Emferisr fmrimuim far p t39iyhedr E and it is Easy EU wr39ify fmm the dima 5llmFI1 139iZ Ei in the rmwinVg table 0 Tetrahe dmn 4 6 4 Cuba 8 12 6 Octahedmn A E 12 3 D cnAderAahedr0n 72H 3U 12 VImsahedrm1 M 3U 20 This furm11la i5 als valid for any iArrgula1r39 pD ly39hedrrUrn as nzing as it is Simplemwhich means that it has nu hUlVesf in it 50 iih t its surface can be defm med cuAnti1139u usly in rm the urfare mi spher e Figure 29 shmvs hm Simple irregular pmlr4hedra for which E F ED 6 E K and w 6 9 5 2 Hnwever E uV1er 5 ftrrrnuia l39I1llE7t be extEnded to V EF2 2p in the case uf a pulyhEdmn with p hnrjles a simple p013rhEdr40m is one fur 75 V7 p mGURI 29 SECOND ORDER LINEAR EQUATIONS 145 I I 1 quot39 139 39 7l quot 39r quotr39j quot it 2J Ar ll 397 39 I 39 1 39rr39 LK t y FGURE30 2 which p 0 Figure 30 illustrates the cases p 1 and p 2 here we haveV EF163216Owhenp1andVEF 24 44 18 S 2 when p 2 The signi cance of these ideas can best be understood by imagining a polyhedron to be a hollow gure with a surface made of thin rubber and in ating it until it becomes smooth We no longer have at faces and straight edges but instead a map on the surface consisting of curved regions their boundaries and points where boundaries meet The number V E F has the same value for all maps on our surface and is called the Euler characteristic of this surface The number p is called the genus of the surface These two numbers and the relation between them given by the equation V E F 2 2p are evidently unchanged when the surface is continuously deformed by stretching or bending Intrinsic geometric properties of this kind which have little connection with the type of geometry concerned with lengths angles and areas are called topological The serious study of such topological properties has greatly increased during the past century and has furnished valuable insights to many branches of mathematics and science The distinction between pure and applied mathematics did not exist in Euler s day and for him the entire physical universe was a convenient object whose diverse phenomena offered scope for his methods of analysis The foundations of classical mechanics had been laid down by Newton but Euler was the principal architect In his treatise of 1736 he was the rst to explicitly introduce the concept of a mass point or particle and he was also the rst to study the acceleration of a particle moving along any curve and to use the notion of a vector in connection with velocity and acceleration His continued successes in mathematical physics were so numerous and his in uence was so pervasive that most of his discoveries are not credited to him at all and are taken for granted by physicists as part of the natural order of things However we do have Euler s equations of motion for the rotation of a rigid body Euler s hydrodynamic equation for the ow of an ideal incompressible uid 22 Proofs of Euler s formula and its extension are given on pp 236240 and 256 259 of R Courant and H Robbins What Is Mathematics Oxford University Press 1941 See also G Polya op cit pp 3543 M6 DiFF39EElEh T lA eeLJsT Iess Euierls law fer the hending ef ielestie iheenss sneli Euier s eritieell ieedl in the meet ef the l mehlilng elf eelumnass Dn severe eleessiens the thread emf his seienltli iel thought led him the ideas his eemempereries were not ready te sslslimi iisIel Fer eslsmpiei he lferesew the phenemeniee ei 1I Ed i3iiiGlFl prelssere wshieh is elreeilsli fer the medem theesy ef the sfishiiity lei stars mere than eeinmr3F heiere Mlsswelll reiseevered it in his ewe wierk eh eleetremsghetislmi Ewes was the ShsZllteilspeslre M 1ilFlaI39i39lErnali svllni s39El5Eil 51 riehijy dies tl il i aQ end inlesheulstihle 0 0 ihrfelst people are sequssihtlecli in semei degree with the Lnsme and reputsltien ef Iseee Niewiein 16 l2a17ZZ ifer his lunlilversel fame as the diseevererl ef the law ef resitetien has ejentinped undimilniisheilds ever the We and s hsif eeinituries sihee his death it is less well krlD V1iiquotri g hew everl that in the imn1ens e sssreep ef vest eeihiievemems he virieslily C39E EEI39El2i mderh plhyisieai seilelnee shed in eernsequvenee has had sh eieeper in uences en the direvcstiehi pf leilvilhsedl life then the rise and fell I1I f estiensl These in e Apesitiern te pjludgel have been unsnimeus in eserk sidlerihg him lime ef the very few SLIKPIBWIIE inlt ellseets the human wee has pquotCldLllCEd N ewtin Wes ihearvrtlz in at farm i7en1 iliy in the siiLiegie eff Weellsiherpe in nen hem Little is knmsn ef his eerlip 339eeris end hndseirgrledieie ate iife st CemThIie gie seernis tel have heee 0u twer dip undisiiepguishelldlil in 1665 an eethrelsk ef the pllegue eeesed the lL1li iquot39s EI iti wS tel elese emgl NEWi l 1 returned he his hwme in the eeuntlrp where he i Er39 vEi Ed lumil leihi Thlerlesi in peers elf rustic sleliiiudile freim age 22 site eIeeti ve pgemiusl iJLlllEiZ ferih in 21 eelcl ef diseevlerhiells unmatched in the hlisierp ef 39lflLliITI I39lJ ltlheughiti the hiinemisl series fer hiegstisre and frsetieneli es penents di erentiei seed integrel eseuiius umiverselli pglrsvlitetieni as the llse5r te the meehegnisrh sari the solar system end the rieselmien elf Ll lIquotlliighi ihie the quoteisuai speielimrnl hp melsnsi elf a prism with its impllilestliernls P ul39 1dE39I S39iEtI1di 7lg leeilers ef the rsinllew and the nieltme ef slight in gen e1raii In p ldf ege he lrerniniseed as feiiews sheet this mii1reeuleus peried ef his 3leeth sin these days I wash in the primes elf my sge fer 3 Fee fufllh f ihl3eimsiieei see z Tr uiesdell quotLeehherd Euler Sepquotreme ilieemeisei lTi 39 x in fH f ET in Eighreee39rhafeeserp i39sihsrei Csse Wessiere ReJserslte iLslni39ws39siip Press IWE Ali151 ihe lhieissemhelr 1933 issue ei hfeeiueeieries Mege39i39irE39 is er heIl39 deseeemi Le Euler see his werls SECOND ORDER LINEAR EQUATIONS 147 invention and minded Mathematicks and Philosophy ie science more than at any time since 2 Newton was always an inward and secretive man and for the most part kept his monumental discoveries to himself He had no itch to publish and most of his great works had to be dragged out of him by the cajolery and persistence of his friends Nevertheless his unique ability was so evident to his teacher Isaac Barrow that in 1669 Barrow resigned his professorship in favor of his pupil an unheard of event in academic life and Newton settled down at Cambridge for the next 27 years His mathematical discoveries were never really published in connected form they became known in a limited way almost by accident through conversations and replies to questions put to him in correspondence He seems to have regarded his mathematics mainly as a fruitful tool for the study of scienti c problems and of comparatively little interest in itself Meanwhile Leibniz in Germany had also invented calculus independ ently and by his active correspondence with the Bernoullis and the later work of Euler leadership in the new analysis passed to the Continent where it remained for 200 years Not much is known about Newton s life at Cambridge in the early years of his professorship but it is certain that optics and the construction of telescopes were among his main interests He experimented with many techniques for grinding lenses using tools which he made himself and about 1670 built the rst re ecting telescope the earliest ancestor of the great instruments in use today at Mount Palomar and throughout the world The pertinence and simplicity of his prismatic analysis of sunlight have always marked this early work as one of the timeless classics of experimental science But this was only the beginning for he went further and further in penetrating the mysteries of light and all his efforts in this direction continued to display experimental genius of the highest order He published some of his discoveries but they were greeted with such contentious stupidity by the leading scientists of the day that he retired back into his shell with a strengthened resolve to work thereafter 24 The full text of this autobiographical statement probably written sometime in the period 1714 1720 is given on pp 291292 of 1 Bernard Cohen Introduction to Newton s Principia Harvard University Press 1971 The present writer owns a photograph of the original document 25 It is interesting to read Newton s correspondence with Leibniz via Oldenburg in 1676 and 1677 see The Correspondence of Isaac Newton Cambridge University Press 19591976 6 volumes so far In Items 165 172 188 and 209 Newton discusses his binomial series but conceals in anagrams his ideas about calculus and differential equations while Leibniz freely reveals his own version of calculus Item 190 is also of considerable interest for in it Newton records what is probably the earliest statement and proof of the Fundamental Theorem of Calculus SECOND ORDER LINEAR EQUATIONS 149 Christopher was little satis ed that he could do it and tho Mr Hooke then promised to show it him I do not yet nd that in that particular he has been as good as his word It seems clear that Halley and Wren considered Hooke s assertions to be merely empty boasts A few months later Halley found an opportunity to visit Newton in Cambridge and put the question to him What would be the curve described by the planets on the supposition that gravity diminishes as the square of the distance Newton answered immedi ately An ellipse Struck with joy and amazement Halley asked him how he knew that Why said Newton I have calculated it Not guessed or surmised or conjectured but calculated Halley wanted to see the calculations at once but Newton was unable to nd the papers It is interesting to speculate on Halley s emotions when he realized that the ageold problem of how the solar system works had at last been solved but that the solver hadn t bothered to tell anybody and had even lost his notes Newton promised to write out the theorems and proofs again and send them to Halley which he did In the course of ful lling his promise he rekindled his own interest in the subject and went on and greatly broadened the scope of his researches In his scienti c efforts Newton somewhat resembled a live volcano with long periods of quiescence punctuated from time to time by massive eruptions of almost superhuman activity The Principia was written in 18 incredible months of total concentration and when it was published in 1687 it was immediately recognized as one of the supreme achievements of the human mind It is still universally considered to be the greatest contribution to science ever made by one man In it he laid down the basic principles of theoretical mechanics and uid dynamics gave the rst mathematical treatment of wave motion deduced Kepler s laws from the inverse square law of gravitation and explained the orbits of comets calculated the masses of the earth the sun and the planets with satellites accounted for the attened shape of the earth and used this to explain the precession of the equinoxes and founded the theory of tides These are only a few of the splendors of this prodigious work The Principia has always been a dif cult book to read for the style has an inhuman quality of icy remoteness which perhaps is appropriate to the 29 Correspondence Item 289 30 For additional details and the sources of our information about these events see Cohen op cit pp 4754 3 A valuable outline of the contents of the Principia is given in Chapter VI of W W Rouse Ball An Essay on Newton s Principia rst published in 1893 reprinted in 1972 by Johnson Reprint Corp New York t D FFERE ilTi L EQ39iJr tTi39Di 35 gt I1deuIquot of the l39lquotlE39Ii l39 iE Aflse the densely packed mathernaties eeansisits arltnest entirely at classical geernetry which was litttle enltisaitieel then and less see nreiwi32 In his dyiniarniesi anti celestial rneehanies Newton iaeihriiiesetl the sisters fer which Cepieirnieius Kepler and Galilee had prepare tl the wiay This sietery was se eerrnplete that the werk ef the greatest seientists in these elcls ever the next two een39turies arntmtntetl rte little rnre than feetrnetes the his eelesisai synthesis It is alse werth remembering in this eenttext that the science at speetireseepy which metre thian any eth er has heen reispnlsiihlle tier erstendinag 3i3tr U39nCFmi C il lcneAsrl edge 39hiejyend the salar system te the unis erse at large had its erigin in Newteni is speetral ianalysis at sunlight Alters the rmirghw surge at genius that went irtte the erieatien ef the Plrineirleia Neiwten again turned iaway tram seienee Hewlelser in a fameus letter the Benxtley in 1692 he eifferied the rst E k d speeulatiens an hms then universe int stars might have ide Feile ptetl eut elf a printerdial featureless el eud eat eesmie dust lt seems rte rue that it the matter at eur S lLlI I and Planets and all ithe rnatter in the Universe was evenly seattefrecl threueheu39t all the heaseris and every partieie has an lf39ll i 139t J grasi ty tawams all the rest same at it wenld eense39ne inte ene mass and sense inter artether se as its tttaite an in nit e ll 1l3aE139 at great ntasses sea39ttered at great dis tanees frant ease tel arietheir trh re ngheut all t39hat in n irt e space And thus tniht the Sun and Fiat stars he termed suppetsirng the matter srer39ei at a ltpleid nratnre This was the heinr1ing at seienitii e eesmellegy and later led 39threugrhr the icleas ef Thentas Wright lan39t Herschel and their sueeessrs tea the elaherate and eensinieiing theery ef the nature and origin let the unirversie pmsided by late twentietl391 eetttttry aist re11emy In I lgl l Newton suffered a sesiere mental illness aeeempaniemzl by sleilnsiiens deep ntelanelhely and tears at perseeutien Ile eemplained that he eenflti nt sleept and said that he rl lli ill his terrmer eensisteniey p O mind He lashetl em with wild aeensatiens in El39lIIClll391g letiters he his friendsi Samuel Pepys an l Jiehrn Leeke rPepys was inferrngted that their friendship was es39er and that Ne witen iweuld see thins ne mere ltZtCl wast ehargeti with trying ta eritangle him with wenten and C being at it The nineteenth een39tnry HIquotltlsl39l philese39pher Whewall has a a39l i39lIl rerrtarl2 ahant this Nt3rl3Il1 lquot sinee itlewtetti has heen able te use gieemetrieal methetls lie the same EHEEIIM fer the iflte urpeses ami as we teat the Priasiipis we feel as when we are in an aneien39t arntenryr where the weapens are at gigantie sizgei and as we ltlHIfIlr at there we mara39el what tttanrier at quotman he was whe eetlllti use as ea weapan what we can svcareeh39 lift as a hItrLienquot 33 Cerrespeaderiee item 398 SECOND ORDER LINEAR EQUATIONS 151 Hobbist a follower of Hobbes ie an atheist and materialist Both men feared for Newton s sanity They responded with careful concern and wise humanity and the crisis passed In 1696 Newton left Cambridge for London to become Warden and soon Master of the Mint and during the remainder of his long life he entered a little into society and even began to enjoy his unique position at the pinnacle of scienti c fame These changes in his interests and surroundings did not reflect any decrease in his unrivaled intellectual powers For example late one afternoon at the end of a hard day at the Mint he learned of a nowfamous problem that the Swiss scientist John Bernoulli had posed as a challenge to the most acute mathematicians of the entire world The problem can be stated as follows Suppose two nails are driven at random into a wall and let the upper nail be connected to the lower by a wire in the shape of a smooth curve What is the shape of the wire down which a bead will slide without friction under the in uence of gravity so as to pass from the upper nail to the lower in the least possible time This is Bernoulli s brachistochrone shortest time problem Newton recognized it at once as a challenge to himself from the Continental mathematicians and in spite of being out of the habit of scienti c thought he summoned his resources and solved it that evening before going to bed His solution was published anonymously and when Bernoulli saw it he wryly remarked I recognize the lion by his claw Of much greater signi cance for science was the publication of his Opticks in 1704 In this book he drew together and extended his early work on light and color As an appendix he added his famous Queries or speculations on areas of science that lay beyond his grasp in the future In part the Queries relate to his lifelong preoccupation with chemistry or alchemy as it was then called He formed many tentative but exceed ingly careful conclusions always founded on experiment about the probable nature of matter and though the testing of his speculations about atoms and even nuclei had to await the re ned experimental work of the late nineteenth and early twentieth centuries he has been proven absolutely correct in the main outlines of his ideas So in this eld of science too in the prodigious reach and accuracy of his scienti c imagination he passed far beyond not only his contemporaries but also many generations of his successors In addition we quote two astonishing remarks from Queries 1 and 30 respectively Do Not Bodies act upon Light at a distance and by their action bend its Rays and Are not 34 Correspondence Items 420 421 and 426 35See S I Vavilov Newton and the Atomic Theory in Newton Tercemenary Celebrations Cambridge University Press 1947 pj i 2iI39FFE RAEHTML EUUATWUNS gressi iiediies s11d5 Ligihit eimwesirtiihile inter heme sinieaithierquot it seerms as eliesr as weridis can he that iwecwiten herve iE 39 jamp 39iliI39i g the griaisitstiienisi hehding hi iighi send the Equ iy I EnCE eii msss and ehgergy iwhich are mime eensreiqueiihiieies ef the iheery of reieiiviityi The tfsrimier piheneamienin was first ehsrewe d during the total seller eeliiipse sf 9 shdi the latter is new hhewn ten uindesiie the ErIv3139 genersn ed by the sun and the starts On ether eeeesiiehsi ss well he seerns is have hnewn in same mysterieus ihihtuiiitise sway far mere than he ewes ever willing er shl ts jusitiifgr as sin this eryptiie sentence in s i ettAer to is friend quot39I1quots plain lie me hy the 39f Ui7 i chew it fi39em theugh I wiii net uinderrtsike quotre prove it the eitihiersiquot3539 iWhsti e seri the illif this 39f Ul 31i P hsse heen it U dLiib IEdil397 dependedi en i lmr a rdi E Er pew39 ersi est eemieni139siieni 0 J asked hms he made his disceivesies he isssiid 0 keep the subject eens1snil3r hemrie me send wait Iii the rst dswningsi sperm lime hy lsiitlie i7139lti J the fiuii iighL This seruhds simple ehieughi but everyeniei with esiperiienee in science of mathematics hnews hewe every diii ieuiit it is he held s problem eentihuwuisily in mind fer more than a few seehcis er a few m inu1esi Ones ettehtieh iagsi the preihleim r epest ediIy slips aiwagy and repea tedsiy has the he dragged hsek an e 7DI39i hf willh Frem eeieeu39ms of wiIhesises Mewtien seems te have heeh eispshie emf sImest39 e ieriiiessi sustsinsed CCi39 CE1Wt H i h his prehiems fer heur s amid days send weeks with seven the hieed hr eessi ensi feed and isieiep sesreleh i iiEIquotfrU39ptil g the siesdr squieeizriiing gri hf mihd In Nieiwitein reeeived sh ieiter from his Jsfeirvdi matheimsticei friend Jnhn Wallis ersmsihing hefws that east 3 eieud ever the rest hf iife Wsiiihg sheet Niiewtenfs esrIy msthemstiesli diseerverizesi Wsiiliisi warned him th t in HeIism reuir39 NiM iiiensiquot are fiaznswn es iquotlLeibniz s Ceieeiiiss Diifferes iieiijsi i siznd he urgejci Newteh its iske steps in pr heci his I puti ii f that time the IeisiiDns VbEEWEEM Ne39w teni Leibniz were stiii eiezrdisi and mUt x 1ig TESg3iECquotIfLii HUWEVEIL WsIiiiis 7s Liieiiters seen eurdiliedi the stmespiheIei and ihiiiatelj the most pi i gE hiiiitieiri and damaging Bf sil seienii e qiusirieilisi the ffameus er ihfemieus Newman LeibniE priiJrity eentre3wersfy ever the ini venti n Uf ieelieulitls it is FLOW well estebilisfhesl that each man desreiieped 0xr sewn fhrm ef esieulius in iepend7entiy of the etheiz thst iN ewten was Firsts iby 3 or l jgesrs 0 III puhiish q ideas end that Leihni s pispeiris hf 634 and 1686 were the earliest publiestiehs h the su hjeeti iHw e39seir what are new pezrieieisedi as siimpiies faetsi where set h esriy39 s eieisr at the time There were eminehs miner rhimihiihgis for years after Wsiiiisi s letters as 3395 FFE P0 ffE quotE I Eem 13993 37 rf339errespenden eej iI eiT1s 4981 and 503 SECOND ORDER LINEAR EQUATIONS 153 storm gathered What began as mild innuendoes rapidly escalated into blunt charges of plagiarism on both sides Egged on by followers anxious to win a reputation under his auspices Newton allowed himself to be drawn into the centre of the fray and once his temper was aroused by accusations of dishonesty his anger was beyond constraint Leibniz s conduct of the controversy was not pleasant and yet it paled beside that of Newton Although he never appeared in public Newton wrote most of the pieces that appeared in his defense publishing them under the names of his young men who never demurred As president of the Royal Society he appointed an impartial committee to investigate the issue secretly wrote the report officially published by the society in 1712 and reviewed it anonymously in the Philosophical Transactions Even Leibniz s death could not allay Newton s wrath and he continued to pursue the enemy beyond the grave The battle with Leibniz the irrepressible need to efface the charge of dishonesty dominated the nal 25 years of Newton s life Almost any paper on any subject from those years is apt to be interrupted by a furious paragraph against the German philosopher as he honed the instruments of his fury ever more keenly All this was bad enough but the disastrous effect of the controversy on British science and mathematics was much more serious It became a matter of patriotic loyalty for the British to use Newton s geometrical methods and clumsy calculus notations and to look down their noses at the upstart work being done on the Continent However Leibniz s analytical methods proved to be far more fruitful and effective and it was his followers who were the moving spirits in the richest period of development in mathematical history What has been called the Great Sulk continued for the British the work of the Bernoullis Euler Lagrange Laplace Gauss and Riemann remained a closed book and British mathematics sank into a coma of impotence and irrelevancy that lasted through most of the eighteenth and nineteenth centuries Newton has often been thought of and described as the ultimate rationalist the embodiment of the Age of Reason His conventional image is that of a worthy but dull absentminded professor in a foolish powdered wig But nothing could be further from the truth This is not the place to discuss or attempt to analyze his psychotic aming rages or his monstrous vengeful hatreds that were unquenched by the death of his enemies and continued at full strength to the end of his own life or the 58 sins he listed in the private confession he wrote in 1662 or his secretiveness and shrinking insecurity or his peculiar relations with 38 Richard S Westfall in the Encyclopaedia Britannica 154 DIFFEREHTML enIUaTInn39a wnmen especially with his mnther whe he thseughrt had abandened P at the age ef And what are we in make of the huesheela ef unpublished III1 ilJ1SCI39ipTS miHei0n5 f wards and thnuaanede nf hnura Inf Ihnngfh that refiieet his secret lifelong studies ef ancient eherennlngy eerily Clheristeian dDetrin e and fhe pmphejciea elf Daniel and SI Jenn Newtenquota desire tn 1mew had leittile in eemmnn with the smug ratienaelism the fight th century 011 the eennatrary it was a farm f desperaten selfpreaervaitinan agaai neet the dark quotforces that he felt pressing in amunede As an nriginal Ihinkeer in aeienee and mathaernaties he was a stnpendcma genaiua whose impact en the wnr391d ean be aeen by EVEfjf e i haunt an a man he was an et139ange in eveery way that normal peeplle can aeareely begin In understand him It is yperheaps mast aeeuratee to think 31 him in medieval terne1se Aae a eoensemated seliltary intnitivei mystic fer whnmn science and mathematics were means f reading the ridaI e nf the unniversen wATquot hre beat e ri is Frank E M39anue a enee enst heMk P anrnI39nquotE nf1iquot5eac N ewminnv Hawan UnivereiI3 Frees I968 CHAPTER 4 QUALITATIVE PROPERTIES OF SOLUTIONS 24 OSCILLATIONS AND THE STURM SEPARATION THEOREM It is natural to feel that a differential equation should be solved and one of the main aims of our work in Chapter 3 was to develop ways of nding explicit solutions of the second order linear equation yquot Pltxgty39 Qltxgty 0 1 Unfortunately however as we have tried to emphasize it is rarely possible to solve this equation in terms of familiar elementary functions This situation leads us to seek wider vistas by formulating the problem at a higher level and to recognize that our real goal is to understand the nature and properties of the solutions of 1 If this goal can be attained by means of elementary formulas for these solutions well and good If not then we try to open up other paths to the same destination In this brief chapter we turn our attention to the problem of learning what we can about the essential characteristics of the solutions of 1 by direct analysis of the equation itself in the absence of formal expressions for these solutions It is surprising how much interesting and useful informa tion can be gained in this way 155 115 i tniIFFEeeH 1e39mL ieeuwrmewe en iiiuetreitim i ef the idea thell Ii i fy ipireiertiee ef the eeietieinie ef e di erer1tiel erquieitieii can be diseeverweci 1 ei udy ing the Lequeitiien itself withedut siemng it in en tre diitienai sense we dieee5s the familiar eqee en Wy Q We knew perfeeetiiy well lthet y11 E eiI1r and E cease ere twe iinearily indeipenrient eaeleitriieinsi ef v that tiiejr are fully determiined by the iniitiei eendiit i enie y1e U y iiD and y3 y U U eznd that the general ee etien is Ayw x e1 39139I7 IF Eiysg lxjii Nreirmieeiiliy we regard 2 e5 eiempiieteiy eeived by these ebeervartieris fer the funietiensi sins and eeex ere eid ifriemille we knew a greet diieel b llt tiherni Hewever eur Lkillii rwi dg ef einx enel ener een be tiheiught ef es en eeeiidiejnii ef hieiieirfygz enri fer the seine ef emphesizin euir present paint efquot view we new pretend tetei ignreneelt ef these femiiiier furieiiens ur purpeee is tee see inrew t39heii39 piriepiertiee een be equeeeedi em ef 2 and the inii1iiai eendiiEiiene P satisfy The einiy teellie we ehieili uee ere quialzitatiive ei gemen Ie and the general priiriieipiee dieeeriibeid in Seetiene 14 and 15 Ae1eiieiidiiiniggii lei xix be dfe ined es the S Ii U iD i ef 2 dequottieri mined by the initial ieieimd1itieniisi3i H end e v if we try tee sketch the grapi39i ef efxi P letting ix increase frern D the initial eemii39Liien5 tell us be steerquot ililfie eunrez at the erigiin and let it rieei with eiepe beginning at 1 Fig From the eqruetien itself we 11evei 3 xi39 3r ED whien tlfie eur ei is ebeve thee t39miii E39F is e negative nunilbei itheti iriereeeee in megn39tLide eye the curve rises Siiinee 539iquotlJf ie the rate ef change ef the sl pe squot x this eiepe deereeases an an iinereieeing rate as the eurve iiqfftei and p must reeeh D et eeme paint 1quot 2 m A5 2 eemiriuee to ineree5ei thee curve fails iewierd thee xe ezziei 3 i2 decreases ell e dEG39e 5iirIg 3tE j end the euWe meseiee me xexis at a peint we een cie ee tie be Since e u1 depends enly en Mfr we see that the 9 betweee 1 E 0 and ex e is eyi39nmetirie ebmit the line 91 re 50 PM Jrfl end P 2 1 E Sil39l39lii3I iergemreint Shewe that the next pertiien ef the eunre is en iirwerted replica ef the Z end en i ndei eiteiiy 1H i F IGUREi 31 QUALITATIVE PROPERTIES OF SOLUTIONS P In order to make further progress it is convenient at this stage to introduce y cx as the solution of 2 determined by the initial conditions cO 1 and c O 0 These conditions tell us Fig 31 that the graph of cx starts at the point 01 and moves to the right with slope beginning at 0 since by equation 2 we know that c x cx the same reasoning as before shows that the curve bends down and crosses the xaxis It is natural to conjecture that the height of the rst arch of sx is 1 that the rst zero of cx is Jr2 etc but to establish these guesses as facts we begin by showing that s x cx and c x sx 3 To prove the rst statement we start by observing that 2 yields y y O or y quot y 0 so the derivative of any solution of 2 is again a solution see Problem 174 Thus s39x and cx are both solutions of 2 and by Theorem 14A it suf ces to show that they have the same values and the same derivatives at x 0 This follows at once from s O 1 cO 1 and s 0 sO 0 c O 0 The second formula in 3 is an immediate consequence of the rst for c x s x sx We now use 3 to prove sx2 cx2 1 4 Since the derivative of the left side of 4 is 2sxcx 2cxsx which is O we see that sx2 cx2 equals a constant and this constant must be 1 because s02 c02 1 It follows at once from 4 that the height of the rst arch of sx is 1 and that the rst zero of cx is 152 This result also enables us to show that sx and cx are linearly independent for their Wronskian is Wl5x CXl 5xC39x CxS39x six2 cx2 1 In much the same waywe can continue and establish the following additional facts sx a sxca cxsa 5 cx a cxca sxsa 6 s2x 2 2sxcx 7 c2x cx2 sx2 8 sx 27 sx 9 cx 27 cx 10 0 r iD1iiT 391 E39l5i39EJquot39quot39Iiigs El3UATIiCi HS The pirssfs are shut dii 1icult and ws ilsssvs sham In this msidssr rsss Prsrhlsm 1 sm sig UiihEi things it is sassy its sss fmm th sibmis rssuiiltsg that the pssitiss ssrsas t1fsx and sxs ssrss rsspsstirsly Is 251 Sm fh ands rsi quot392s sraquot2 st sf2 Est a H P ThEialf arcs twu imam ipsiintss is be made shmm ths sbCIquotas iEiEwi 39i 39Hii First ws hisss sisstrisctisd slimsst ssvsw signi cant ptrsgpsrty sf the fiumszti ns sin is and ms 12 fmm isqsustii h 2 by this msrthsrds 0fi iI 1FsrE E i i39 Iq39 E E mE signs wistihmut using any prim knswisdgs sf tirignnomistryi Ssssnidl the sssls wish dii use srsnsiisissd chie y 0 W mnssisiiisty srgumsms iirws lsiing X sign and msgnsitudis sf the ssicsnd dsrsissitisvs and the basis gpmpisrLissi sf Ilihssxr squis tismis set forth P Ssstisns M snd 15 I It gsss withsmt ssjyin that Q sf the shims pmsrwtissi sf sin I sn ssrss srs pssulsisr ma thsss fuinstiisns slvuniss Nsssrthsisiss the ssrmssl fsstuss sf their bshissii0sr ths fast that K nssiiilssts in such a msrmsr that thissir zerss are disrtinct and Dusur sih srshstsiyacssni his sgsnisrsslissidi far beyond thsss psrtiu1sr f umtisinrss This fsiliswing sssulst in this diirsstii h is sslllsrd the Ssurm sspsrarisn rhisisrsm1 Thsnr39em D if 1JrI snd y1fx39 ssis Ems iiinsarij independent ssi39ssquotisss sf s 39 P Exls39 Q sy 0 shsn the ssrss of rhsss fwss srins Iquot E distinct ssssd sissass si39Esr39nisssiy siss rihs ssrtss that yj x ssnsishss ssscrhw smfs hsrwssn sissy hm SME39EEhfUgtE39 ssms sf ysIjss39 and ssinssrssiy Prussf Thsh srgumssnt rssts primarily sh this fsss ss s ths lsrsrnss in Ssstisri 15 tihats sinss 39 am y are iinssri339 insdlispssid ssnI tshsirr Wf ShiEI Mquotr Uquot1iquotsquot Trix 39 F2 iyiiii5 disss nut sssishi sud shsrsfsrssi39nss it is ssntinuusus must hihsvs ssnstsnI siigsi First is is sissy ts sass that yi sud ss11ssst hisss s suznmimsn ssrs fszr P shsy day shsri flhn Wrm1siltisn will ssnissh as thst paints s 39hrish is impsssiisbis We ssw rssssusns thi li IQ ssnd s2 BITE s11I1ssssiss ssrss sf gal esnadi shnsw 111 st jaw 1 i391iEi1EE hstwssn thsss psinss Tlss Wmnsliisin sIssrl3a39 rsdmssis this yiixy js Ass rs smrl J53 so hsth fsmtmrs yyisli and y rs are asst st sssh sf silsss pQ39inKtEiu39i F1srt hs ssrs39moss irgs i sud mus39t hsss tsppsrsits ssignsi hssssls ya insr sssiing st I1 is must be dissrsissinsg st xi sud siss vsrssi Sinus the iJssrq39uss ChaI39iES F39rsn sis Emrm i esi ji was s Swiss misthsmstisisn srhu spElm 39IGS lI sf his life in LP ssis FL I39 3 rims has wss I41mT is ths sis Ehvgiis ismi39 snd sfssr ihsidihg ssssrs srthstr psnsisi ns ht st lssl ssssssdTsd fP sis s n in the Chair Di Msshsnics ssi ihs Ssrhrsnns His rsisih 1ssssls was ailtans in w hst is nsw ssMsd ths iisurrm IisiwiHs thss 39 inf differssrtisfl sq L1s39ti rns which has hssh sf s39Issdiiylt39 iiinsssssing iiiriigqpusrtsnss sssr sinss in Issith puss n1sshcmsssiss sud msthsms39Liisl physiss QUALITATIVE PROPERTIES or SOLUTIONS 159 Wronskian has constant sign yx and yx2 must also have opposite signs and therefore by continuity yx must vanish at some point between x and x2 Note that y cannot vanish more than once between x and x2 for if it does then the same argument shows that y2 must vanish between these zeros of y which contradicts the original assumption that x and x2 are successive zeros of y2 The convexity arguments given above in connection with the equation y y 0 make it clear that in discussing the oscillation of solutions it is convenient to deal with equations in which the rst derivative term is missing We now show that any equation of the form yquot Pxy Qxy 0 11 uquot qxu 392 0 12 by a simple change of the dependent variable It is customary to refer to 11 as the standard form and to 12 as the normal form of a homogeneous second order linear equation To write 11 in normal form we put yx 4 uxvx so that y uv u v and yquot uv 2u v uquotv When these expressions are substituted in 11 we obtain can be written as vuquot 2v Pv39u v Pv Q39uu O 13 On setting the coefficient of u equal to zero and solving we nd that U e P quot 14 reduces 13 to the normal form 12 with 1 A 1 6115 tQX ZPX2 39 1 00 15 Since vx as given by 14 never vanishes the above transformation of 11 into 12 has no effect whatever on the zeros of solutions and therefore leaves unaltered the oscillation phenomena which are the objects of our present interest We next show that if qx in 12 is a negative function then the solutions of this equation do not oscillate at all Theorem B If qx lt O and if ux is a nontrivial solution of u qxu 0 then ux has at most one zero Proof Let x0 be a zero of ux so that ux0 0 Since ux is nontrivial ie is not identically zero Theorem 14A implies that u x0 at O For the sake of concreteness we now assume that u x gt 0 so that ux is positive over some interval to the right of x0 Since qx lt O u x qxux is a positive function on the same interval This implies that the slope u x is an increasing function so ux cannot have a zero to the right of x0 and in the same way it has none to the left of xn A similar argument holds when u x lt 0 so ux has either no zeros at all or only one and the proof is complete y titIF39F39EEEtI t tvtLt EDiLf t It39 N5 u 32 Stiqntze tmr itnterest it in the tSC iquot t39i1D at 5lu ti0tnt5 this resultt Iteadst us tat ctm ne tmr FStf39tttl39 at 12 tm ttne spe ciat case in 39Whu39ith q is at nsittiw f 1JIH39C Uquot Even in this EH53 hmwmrm it is mt t 1Et3ES5xariiJtt true 39Il aEt 50Jltttia ns wtitfl useillatet TU get an idea at what is inv0tvetdt let ttfx th 3 nt t39Fitfi l solutinn Dif V122 with q x 22 0 If we CUtSid39E39I at prrttimn at the g139aph Ht 1 t396 the x 3Jtis z than tt 3f39 0 is ttegativ g S0 the graph t5 Crzrncatte duwnt and the slope Et39l139 is daetc39teasitntg If this slimpes avset bemamas tnegativa hen the curve P1f3lif 3I quot trusses the xaaxiss stnmewhere to tha right and we get Ell tetra fur ux We krmw that this happtens wthem qtlftt i cr ntttantt The atttett39ntattivre is that tatlthnugth u x t tdecreasest it ntever reachast EEEG and the cturve mntitnutes ta trtise as in the utpper part nf k It is ft a n bh clear ft Dt1quot1 these temartks that t tIlt will htatvtc zeros as J incttretasag wthetntavetr qJ dogs nut derttetatstt two 1g luadts tts to H1E ntezttt thtGram Tthearemt p c Let tttrV its any nttrttrtt Lit taft39ttt739Dn39 Hf tt j39xItt 0 WhE 3939Z qt f w i far aH 1 Ifquot I am at 163 I th w H tjI tzat t39rt n tE y mtmyquot Z laEquotF1f35 tut the pm5Kttt39tht39 xazztit Pf f Assumtt ttlte Gt 39lI a tquoty nramtl4y that tt J vanissfhtes at must 3 nite ntumher of times frat D c i gZ so that at J it I I At 2 1 tEa39t5tiS witth tthxe prutmpviltrt1r that ttI39 U 0 V tim all x S R We may CiEELEM 5uppn5te with Du t any tots of g tttir aItt jg tttheat t 1t E El f t all 1 I 135 5intt ulfytj can ht ttttplac d by ngafIivE tnecestsary Our p39L1tp39tI 39SE is to nntradut 1 JsE assutmptti tn lay ht1witn that tta p nvegatttiwe snmewt1t3re tn that right Df xFf 39r5 the QUALITATIVE PROPERTIES OF SOLUTIONS 161 above remarks this will imply that ux has a zero to the right of x0 If we put u39x ux for x 2 x0 then a simple calculation shows that vx v x 6106 vx2 and on integrating this from xo to x where x gt x0 we get vx vx0 J qx dx I vx2 dx 0 10 We now use 16 to conclude that vx is positive if x is taken large enough This shows that ux and u x have opposite signs if x is suf ciently large so u x is negative and the proof is complete PROBLEMS 1 Prove formulas 5 to 110 by arguments consistent with the spirit of the preceding discussion Show that the zeros of the functions a sin x b cosx and c sin 2 d cosx are distinct and occur alternately whenever ad bc O Find the normal form of Bessel s equation xflyn xyr x2 O and use it to show that every nontrivial solution has in nitely many positive zeros The hypothesis of Theorem C is false for the Euler equation y kx2y 0 but the conclusion is sometimes true and sometimes false depending on the magnitude of the positive constant k Show that every nontrivial solution has an in nite number of positive zeros if k gt 14 and only a nite number if k 5 14 25 THE STURM COMPARISON THEOREM In this section we continue our study of the oscillation behavior of nontrivial solutions of the differential equation yquot 6106 0 1 where qx is a positive function We begin with a theorem that rules out the possibility of in nitely many oscillations on closed intervals Theorem A Let yx be a nontrivial solution of equation p on a closed interval ab Then yx has at most a nite number of zeros in this interval 162 EA1FFEemrrIeL EDU TIe HS Preef We assume the eemtrary neemeEy tI1etyx has an ein nite nurmbezr ef zeres in eb It fewewe ff f thie the1i theere exierl in eb39 e peim 3 end e eee39ueenee of eejreas en i x such telnet ex 2 em Since yx is eentrinueuse end d ieltEferemie bie eat en we have Mrue Tim vx U quotquot quoti 1 Iii and ff1quot E x ylixul hm A t39 In 0 By Theenrem 14aeA these 5te emen1IS iemply that x is the triviejl 5 hJtien erf 1 and this eeen39lerediteetiem cermpetee the epmuf We new reeea trhegt the Sturem separetie n theetrem tells us that the meters ef any two nentriviele eeluetiens ef 0 either eeineeide er ereeur eiternete39l3r iileepieending en ewhethere these isat utieernes ere Eicneerly iepemjleeent er indeependkeente Thus ail 5elutie ns ef IE1 esei ete er h essentieaHy the semee rapidity in ethe sense thcet D11 3 given interval the number of gems mf any 5eF lutieen eeinneAt di Ve139 by mere than ene from u neeeeemebeer of BEIGE ef any ether 5ztrluetiuJn On thre Dtheer hand it is clear 39tha39t39 s lutiens ef p quot 432 0 v eeeiileete mere repidiy e1thet p n hev39e more eeree than selutiene of W ye 0 3 fear me zeree emf e selutziein ef 2 sucih as y sin m 5 are eniy helef as far apart as the zeros ef e selutin y simax en 3 The efee ewing IquotESu139t which is kneewen as the Srerm emperi3er1 erheereeml sihewse this beheevieeer39 is typviee in the sense that the sefiutiens Df IE1 39 53Cill tg more rapeidly when qx E5 ienerea5eed Theorem P Ler eerie exj be ran39r eJieE39 3eE7t I 31me elf 9quot eierlav 0 end 2 rI rE Age 3 weIere er Y er1d erfx are pewiiiue funEff H3 such Fmr q n 1 rt39 Wren yx Uen eiee ea m ea e39r anee between any we 3ueeefv e 2 EJquotquot739r2J5 elfquot zfx Q Preejquot Letx end 2 he eueeee5ive zerere esf erlt 5e39Iha1 ex1 ze xgj D and er deeee new 39JH I1iSIquotl an the pE39 ei terwle xee We eeee mee that yIIr E We thus infernenee we e5e the eE eerzer WeE39en7re3e r1e39erem deafquot e Ie39varmed eeleulus whiequoth expweesee eee ef the heeie tesjpelegiaeeal prepertiee eIE the real numbeer j3r5 temJ QUALITATIVE PROPERTIES or SOLUTIONS 163 does not vanish on xx2 and prove the theorem by deducing a contradiction It is clear that no loss of generality is involved in supposing that both yx and zx are positive on xx2 for either function can be replaced by its negative if necessary If we emphasize that the Wronslcian W042 yxz39x zxy x is a function of x by writing it Wx then II If 3 W 39rz Z39qy 4 quot ryZ gt 0 on xx2 We now integrateboth sides of this inequality from xi to x2 and obtain Wx2 Wx gt 0 or Wx2 gt Wx However the Wronskian reduces to yxz x at x and X2 so Wx E 0 and Wx2 S O which is the desired contradiction It follows from this theorem that if we have qx gt k2 gt O in equation 1 then any solution must vanish between any two successive zeros of a solution yx sin kx x0 of the equation y kzy O and therefore must vanish in any interval of length Jrk For example if we consider Bessel s equation in normal form 1 4p2 1 u0 u 4x2 and compare this with it u 0 then we at once have the next theorem Theorem C Let ypx be a nontrivial solution of Bessel s equation on the positive xaxis If 0 S p lt 12 then every interval of length TE contains at least one zero of yx if p 12 then the distance between successive zeros of yx is exactly Jr and if p gt 12 then every interval of length It contains at most one zero of ya x Bessel s equation is of considerable importance in mathematical physics The oscillation properties of its solutions expressed in Theorem C and also in Problem 243 and Problem 1 below are clearly of fundamental signi cance for understanding the nature of these solutions In Chapter 8 we shall devote a good deal of effort to nding explicit solutions for Bessel s equation in terms of power series However these P mFFEREhTI aLEEILrA39T39i1mH5 series E 1lLEi39i nS are awiltward tnzmis I0 try In use in Studyring uv5iiAIatiDn pm39upriertiEs and it is a gram cUnvEni enEi in Int abie In turn In quaIi tatihwE mas0mVingi 04fV the kind di5muss 3 sin this uhaptar PROBLEMS 1 Let J5 P 7 xi be sure55iwe pm5itive EEIEIS mi 31 ntrnrriviaI sntutim1 y rA nf 1e2s5e4395 erqua mnm 3 If K 5 L J M2 5hlIJ39W that J52 I1 is less Ihan If and apprg acl1es Jr as 2 as b If p in 1fE shw that 4 e 13 is greater than 1 and app rnau391a Jr as f K 0SG uH If yFr is a mrn39trivial s IruIimm If y G L saftmw that yIr Aims an in rmi391e numher nf pLsiiiIv arr ls if qixi 9 kr39tE fatr Smme R 11 LM and nly 1 39 niI iE num7baEr if q 39 142 Emery n rr139triviaE 5uA1uIi0n mf 3quot 5in1x 1 G has an in rmtite num ber mrf pn5ii39we germs Fnr39mVufat39e and gprmre 3 tfhearem that i4mIudes this statamen as a slpeci l 1235E CHAPTER 5 POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS 26 INTRODUCTION A REVIEW OF POWER SERIES Most of the speci c functions encountered in elementary analysis belong to a class known as the elementary functions In order to describe this class we begin by recalling that an algebraic function is a polynomial a rational function or more generally any function y f x that satis es an equation of the form Pxy Pn1xyquotquot P1xy Pox 0 where each Px is a polynomial The elementary functions consist of the algebraic functions the elementary transcendental or nonalgebraic functions occurring in calculus ie the trigonometric inverse trigono metric exponential and logarithmic functions and all others that can be constructed from these by adding subtracting multiplying dividing or forming a function of a function Thus xelquot tan 1 x2 3 tan A A at y sinx cos2x Vlogx is an elementary function 165 39l6l i ht FFE1i7 hETis i EtQUA l IDT439S Etezyehtl the elermetntettry ftJ t2tiUnS lie the higher Itremeentdentttet ftmett hh5Jt er as they are efterh eellet xthe tspteeiet funettteettt Shtee the htuegihning ef the eighteehtht E t tyi math hLtndt EE15 ef spetetet funetiehe heme heeh etetnstid7 ered euf etienttbr intteretting ht itmpertent te mE fi i smite degree ef SE39Udy 1I quot39 G5t evf these are elmest eempletely fergetlteht but semem such as the gemthe tfmtetiteh the Riemenh zeta ftmetittnt the e iptiC t39unetiene endt these that eetnttitnue te he tueiefull in meuthentetieet tth3reiee Vh fte gehemtted extensive theeriest And erneng Iheee eh few E11fE eh rich in meetning and itn t1xenee that the mete thietettty ef ent3r erte ef them W tlld fill a llerge h eek The etd ef eupeeeiel tfutnetttene was tetutttthetetd with tenttthueieeti et dev etiettt hey htetw hf the greatest meth emettieierte ef the eiqghteentth and mneteenth eetntturie5hy Euler GattS theI Jeeeht Weierstteets Rie merm Hemtittet end Peineer 3J iIquottg etihers But ttestee tethtethge with the tirhee and twee meet I h EFW IiEi 5 prefer te etudy tettge eteeeee ef 39funetierw eehthtueue ftmettienec ithtegtehlte funetiene ete itnsteed ef euatstenthng individuele NEVEiT 7ltE l EE E tihete ere etill rneny whet fever htegtepthy ea39etr eeetttelegy and e hetentctedj til39E 39EmEt1t ref analysis earthe t hegieet either Ki w i Specie f unetiehe veer tether widely with teepeett tea thtetr etigtttjt ttetttrte and ap pt1iEMi I39t5 I3Iew evert ette htirige gtre39utp with e eeheiderehlet degree ef tunttty eetheiette ref these thet arise es eehtttietne etf eetetend ettier hheet i1lz TEti39 I1I ii77tJ eqtletiehett hhtenty ef these nd applieetients in eehnee tit Jt39I with the perttiel differetntiel eq39LIe1iehe erf trt39IJthretmettieet pthyeitee 391 hey ate else tm39pDl39E L threugh the tthtetety eat evrtxhegenei EiEpat39lSiCt S es the main hhiettertieel eeutree ht htteegt enetytete wlhieh Thee pteyetd e eeantrtitt rete in shaping much elf tTquot39I IUdnE1rI t pure metzfhemtetitee Let Me try to U t39IiIElEFEvI d t ie getttet e l W33 Lhew these fmllli U39t1 nA S et iee It wit he teeettsed that if we twtsht t eehre the simple E39qlJaIitT11 then the ffemtiiiet ftJt tlT39i HE sine and y ettetsx are elreetdtgy tatreHehthle fer tthsie putpeee frem lEII1E tt E1Tquot eetetttus The eiet1ttetiteth with tespetzt tie the etqutetietil tt39j 39 y ey 4 quite diatthetrehtttt feet this Eqj39L1 i 1i39iquott eertnett he eehreti ht tetrme ef ElE39f Ital 5quot fL1 n E Wi 1 SW5 As ea mettet hf fetrt there pR net ltittewh type ef I The reedegr twh e wiehee tea term an ittttttreeeieh ef the ettvetttt eat this part ef enetyeie weushdi do well he htitjh thre ttgh the three veiunttee ef Ht39gher T39ntrteeendehte Ftmett39ehtt Etdeiyi letT Me GtewhH til New Teeth 39ll953 1 JS539 POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS 167 second order linear equation apart from those with constant coef cients and equations reducible to these by changes of the inde pendent variable which can be solved in terms of elementary functions In Chapter 4 we found that certain general properties of the solutions of such an equation can often be established without solving the equation at all But if a particular equation of this kind seems important enough to demand some sort of explicit solution what can we do The approach we develop in this chapter is to solve it in terms of power series and to use these series to de ne new special functions We then investigate the properties of these functions by means of their series expansions If we succeed in learning enough about them then they attain the status of familiar functions and can be used as tools for studying the problem that gave rise to the original differential equation Needless to say this program is easier to describe thanto carry out and is worthwhile only in the case of functions with a variety of signi cant applications It is clear from the above remarks that we will be using power series extensively throughout this chapter We take it for granted that most readers are reasonably well acquainted with these series from an earlier course in calculus Nevertheless for the bene t of those whose familiarity with this topic may have faded slightly we present a brief review of the main facts T A An in nite series of the form 2 axquot an 51 azxz 3 n0 is called a power series in x The series I Z anx X0quot 00 alx 950 020 3Cu2 39 4 n is a power series in x x0 and is somewhat more general than 3 However 4 can always be reduced to 3 by replacing x x0 by x which is merely a translation of the coordinate system so for the most part we shall con ne our discussion to power series of the form 3 B The series 3 is said to converge at a point x if the limit lim 2 ax m gt r1 exists and in this case the sum of the series is the value of this limit It is obvious that 3 always converges at the point x 0 With respect to the arrangement of their points of convergence all power series in x fall into one or another or three major categories These are typi ed by the POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS 169 yield the formula 0 R lim npoo anl if this limit exists we put R 00 if aaH gt 00 Regardless of whether this formula can be used or not it is known that R always exists and if R is nite and nonzero then it determines an interval of convergence R lt x lt R such that inside the interval the series con verges and outside the interval it diverges A power series may or may not converge at either endpoint of its interval of convergence C Suppose that 3 converges for x lt R with R gt 0 and denote its sum by fx 2axquotaa1Xa2x2 quot39 n Then fx is automatically continuous and has derivatives of all orders for x lt R Also the series can be differentiated termwise in the sense that f x 2 naxquotquot a Zazx 3a3x2 nl X f x Z nn 1axquotquot2 2a2 3 2a3x n2 and so on and each of the resulting series converges for x lt R These successive differentiated series yield the following basic formula linking the an to fx and its derivatives I f quot 0 n 9 I1 Furthermore it is often useful to know that the series 8 can be integrated termwise provided the limits of integration lie inside the interval of convergence If we have a second power series in x that converges to a function gx for x lt R so that gx 2 bx b bx by 10 n then 8 and 10 can be added or subtracted termwise fx i gx S a i bxquot an i b a l bx n ITW JJFFfE REttTtAL EtJv39LIm39tUHts Thgay can l tit be tml1I ipHEd as if they were JUlFI IU 1i5 l5 in the sense tl39l t ttxm E m tn EH wtherea L anb1 V 1b 2 If it happens that h tlfit seritest 39EtttttuErg fE to w S m t fLtnEI iD t SE ttmtt r grx f 1quot xlt ii R IhIt lr39t ftZLIquotftI39t Ul 1 implies that tthey mustt have the sarner EIf139Ej iv nES an 0 2 1 b 0 In particultar if fix Q fur txnj 1 than an U at 0 0 i 0s Let x Tb a tcnntinutrrus tfutnctittn that has d rivtattivets of an nrd rs ft39lIf39 Ix ii J wittht 3 0 Can fllfx be t vEpI SEt7tt39tfEd by a pnwet tserties It we use 9 tan da rte the am than it is natural tn hnpe that hie tEE39p I13i fl Em rtmx 4 2tT2 M t z ntZI will thcatct thr gh U39t thet igtttertraat 0 is mitten true but tIl 39fCtI39tU t t hi it is somettitmets thllsa One way at iinquot39tt39IEEt t iig Ii g the validity tjt this axpansiun for at 5pBci 1c pt ilil I in the intterttalt P tn use Tayfatrt f rrrtt P at where the remainder Rx given by i39139ltI x r quotquot1 tart SEFFIIE point 1 btatween D anttt 1 Tu verify 11 ztt 5u ice to Show tttat Rnfxtv U as rt r Y By I I1Eat39tE Gf this ptr ntedt1te it i quite etasiyt ta nb taitn the ftft vUWi g famitiztt E2it panSiiD13915 39wh39iE11 are valid f if all t xk Rf 0 xri X2 X3 E c I D Kl 41 It I 2 3 A T 214I 3 v nj r i O l m P0 k 5 39I V M mm 1 I 3 T 5 t M an F x2r39l39 X2quot l A 1 2 E1 J 4 t H u A 14 33911 witll TIE tJSE39f1HJ tazttm tn ttttt tlti thttt ti t tm ht wztittttn in twtgt Et tttittaent ftmttsz Jug P fr H1 hr K J 5 r b u i hitquot J quot1 ll quot15 POWER seams SOLUTIONS AND SPECIAL FUNCTIONS 171 If a speci c convergent power series is given to us how can we recognize the function that is its sum In general it is impossible to do this for very few power series have sums that are familiar elementary functions E A function f x with the property that a power series expansion of the form fx 2 ax X0quot 15 n is valid in some neighborhood of the point x0 is said to be analytic at xo In this case the a are necessarily given by fnx0 n and 15 is called the Taylor series of fx at x0 Thus 12 13 and 14 tell us that ex sin x and cosx are analytic at x0 0 and the given series are the Taylor series of these functions at this point Most questions about analyticity can be answered by means of the following facts I J 1 Polynomials and the functions equot sin x and cosx are analytic at all points 2 If fx and gx are analytic at x0 then fx gx fxgx and f xgx if gx0 9 O are also analytic at 10 3 Iffx is analytic at x0 and fquot x is a continuous inverse then f x is analytic at f X0 if f x0 95 0 4 If gx is analytic at x0 and fx is analytic at gx0 then fgx is analytic at x0 5 The sum of a power series is analytic at all points inside the interval of convergence Some of these statements are quite easy to prove by elementary methods but others are not Generally speaking the behavior of analytic functions can be fully understood only in the broader context of the theory of functions of a complex variable PROBLEMS 1 Use the ratio test to verify that R 0 R 00 and R 1 for the series 5 6 and 7 2 Ifp is not zero or a positive integerpshow that the series i pp 1p 2 i p rt 1x nl n converges for x lt 1 and diverges for Lt gt 1 POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS P in earlier chapters and also that solutions for equations of this kind can often be found in terms of power series Our purpose in this section is to explain the procedure by showing how it works in the case of rst order equations that are easy to solve by elementary methods As our rst example we consider the equation y y 1 We assume that this equation has a power series solution of the form yaoaxa2x2axquot 2 that converges for x lt R with R gt O that is we assume that 1 has a solution that is analytic at the origin A power series can be differentiated term by term in its interval of convergence so y a12a2x3a3x2n1axquot 3 Since y y the series 2 and 3 must have the same coef cients a1 do 2a2a1 3a3a2n1a1a These equations enable us to express each an in terms of a0 01 do 02 an 00 01 00 02 quot quot39 quot39 quotquot quot 39 a quotquot 2 2 3 3 23 quot n When these coefficients are inserted in 2 we obtain our power series solution ya1xJCr 3 r 4 2 3 n where no condition is imposed on ao It is essential to understand that so far this solution is only tentative because we have no guarantee that 1 actually has a power series solution of the form 2 The above argument shows only that if 1 has such a solution then that solution must be 4 However it follows at once from the ratio test that the series in 4 converges for all x so the termbyterm differentiation is valid and 4 really is a solution of 1 In this case we can easily recognize the series in 4 as the power series expansion of ex so 4 can be written as y age Needless to say we can get this solution directly from 1 by separating variables and integrating Nevertheless it is important to realize that 4 would still be a perfectly respectable solution even if 1 were unsolvable by elementary methods and the series in 4 could not be recognized as the expansion of a familiar function This example suggests a useful method for obtaining the power series expansion of a given function nd the differential equation satis ed by the function and then solve this equation by power series POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS o solution we have 1 1xquot1px 2 T 2x2 pp139pn1 M x 9 for x lt 1 This expansion is called the binomial series and generalizes the binomial theorem to the case of an arbitrary exponent PROBLEMS 1 Consider the following differential equations 3 y Zxy b y y 1 In each case nd a power series solution of the form 2 axquot try to recognize the resulting series as the expansion of a familiar function and verify your conclusion by solving the equation directly 2 Consider the following differential equations a xy y b xzy y In each case nd a power series solution of the form 2 axquot solve the equation directly and explain any discrepancies that arise 3 Express sin x in the form of a power series 2 axquot by solving y 1 x2quotquot 2 in two ways Hint Remember the binomial series Use this result to obtain the formula 7l391111393113933951 6 2 2 323 24 52 246 72quot 4 The differential equations considered in the text and preceding problems are all linear The equation y 1y 3 As the reader will recall from elementary algebra the binomial theorem states that if n is a positive integer then nn 1 nn1n 1 2 x2 kg xquot lt1 xquot 2 Zx 1xquot1nx xquot More concisely where the binomial coe icient is de ned by n n nn12n k1 k 39kn 1 quot kl 39 175 ntFFEREtrnmL iE UdrTtIt JII39u39 Et39El nnrtiinctar and it is teas to see tdiracttty that i 2 tan is IME p I139I39I39Ci39ii3L ampEI39 5 u tiDn EDI whirzltl d as 0 Sh w that ttanx I 1325 Exj E m k k k 15 by a53umintg a sututiunt far equati n in in the ft rrrr1 0f 3 power seriees k H51 and ndirtg the 1 in twat ways la by the g of the examples in thE text Ifncatet jpvarticuttarty hm the rm3nlinearityt Hf the eqtuatimt mmptli cates the frrntu1astj In by di7 EIquotBI39ItiEL g ttquattitont ii F p t dih to obtain 1quot 2yyquot1 W E Zyt Etta TJE t 9 and using the f Lf 1U an f quot fDn 5 SizeEve the eqttatittn J I Jan HA0 0 by each f thc l I39lB Il391 dE suggested in P tcrbtltamt U What familiar 39futnttion dues the mttualting Kserties rBPt E SBt39I39t 4P 3939fiJlIr EU EtuSti 0 Q sawing the equattit nt directly as E ttt DI dEJT linear eqtmtatti nt P O P O E We tmw tum DUI aHE I iUH tn tlw ganeral hutm ngen wust secntnd 0 rder linear eqtuatti nn W P J 4 tQ 1y 0 13 As we lttnnw it is ctcasi allty PKUSSibl t ta salve such an etquatinit in terms of fami ti r eIementtarr fm Eti nEiI This is ttrtm fr inttancm whent P39Il and Qa x are Ct39 JI tI t I1t5j and in a few other EH5E5 as well Fm the rmtrstt arttt htttwev etg the eqtuattittns turf this typ thtavintg the grezaatest simi tvcatnce int both um and tapplitetd mathematics are bty nd the f h of eletmertmt t tary metlt ds aml can null be suited by means of pawer tetrtiets The ttzentml tfact abmtt equattti n l is that 739lE Vethavintr at its soEtutitms nztart a paint Jan tdTeptend5t mt thus bshavtinvr nf its E E39 t iEn1 fI In39 ti S PliJtT and Qt near this paint In tthits sectin we3 CD 1 E tiirlL1 l SaEIJ394 1352 tn HIE 2315 m wthticht P r and QM am WE behavedquot in the sense Df itlg angalyttitct at gm which mtanrs that each a paws Sarita expansion vatlitd in n eigthtbtor ht0od at this p0aiI 39t39 Iln this case xn is calied an thiin T Wpmm of equatiDrr1 1 and pm But that ever 5nmti0n at Ih equati nt is 321150 analyti at 1iS paint In thcr wrds the anaIyti ctitty aft the E E ljE39iEi I S at 1 at tat certaittn pmigm implies mitt its S 1Uti S am 3150 a l fti them Any paiitntt that is tmrttt an nrdtinatyt ip ittt of 1 is trallad a sttngtttattr pt1in tt POWER SERIES soumows AND SPECIAL FUNCTIONS 177 We shall prove the statement made in the above paragraph but rst we consider some illustrative examples In the case of the familiar equation yr y 2 0 the coe icient functions are Px 0 and Qx 1 These functions are analytic at all points so we seek a solution of the form ya0axa2x2jaxquot 3 Differentiating 3 yields y a Zazx 3a3x2 n 1ax 4 and i y Zaz 23a3x 34a4x2 n 1n 392a2xquot39 39 5 If we substitute 5 and 3 into 2 and add the two series term by term we get 2a2 a0 2 3a3 ax 3 4a4 a2x2 4 515 a3x3 n 1n 2a2axquot 0 and equating to zero the coefficients of successive powers of x gives 2a2a0O 23a3aO 34a4a20 45a5a30n 1n 2a2a 0 By means of these equations we can express an in terms of an or 1 according as n is even or odd a 00 a an a 02 00 2 l 3 j 4 quotquot 2 23 34 234 613 a a5 39 1 With these coef cients 3 becomes 1 0 1 0 y 1quot quot x2quot quot3n 39 T3 x2 x4 x3 x5 a01 5Z1 ax 3T p Let y1x and y2x denote the two series in parentheses We have shown formally that 6 satis es 2 for any two constants an and a In particular by choosing an 1 and a 0 we see that y satis es this equation and the choice a0 O and a 1 shows that y also satis es the equation Just as in the examples of the previous section the only 173 ettFeeeHrtsL EQ1U39equotF IDN rtemshfting issue eeneetnst the eetwetgeihee est the tee senies dte nirng y and Ry3 But the tia test shows at ettee eeeh ef these seties ettd therefet et the series 6 eetwet ges fee all he sees Prehlem It fitellews that all the Upere ti e S perfemted en 3 are tegitimatte se 91 is at vslid settmtieh of 2 es teppesed te e metrtely terms selutirent Furt hermete eyes and J are li t ifw in dependleh39t since it is ehtwieus that 11either series is a ICDI1Sitt31 t1t multiple ef the ether We thereferhe see that V is the gettetel seh1tiem sf 2 and that any perti euIet sehttiett is ehtsitted by tspeerifyithg the sehsest ef ea and yquot39 5 tel In the sleeve example the tee series in peret1th eses PETE easily teeegnizs hle es the E tfpt I1Si S ef eessc and 1 se 6 Em Izze W39t39 tE in the farms y en sesst 4 1SiI1I39 i hetttetrally this eetnehtsien teeutdt have been fJI39E5EET1 in the h egienihne since 2 is a very simple eqtutetien whose selutiens erue p ff t f familiar te us Hewevert this result sheu1d e regsttted see my El htcky eeeidettt fer mest series setutiems feuhd in this way are qtlit impessihIe ts ihdentify and IEPFESEn PtTE ti LlSI39 U ik Dw ftIamptnEt i t S Asst en illtusttatietn ef this IquotE t39t1a1Fk II we use the same pIDvEEdtlIEI te seltve Leg etm t39re ts equa39I tert 1 IEW E 2e ms Us U T where p is e eenstent ilt is sleet that the eee ieient fllIE5l39E3IJ2iI lS I 1 I ere snslytie st the erigin The ettigin is thetefere an etdinsry point and we expect s t5Q139t1JtiDiTt ef the term y E ewe Sines 39quot E e 1 ext we get the fDil39I ewing espensiens fer the indjhritdutel terms en the left side f equeti en F s y n I1n xEt F 1t H xrtj 2 Qn1lE he lites and T Ms 0bz lily Seth 1 Pt39xH39 By equstthien f39r39 the sums ef these series is retquited te he 2611 se the eee ieiehttt of 31quot must be eete fer every ten I lxn 2e1E rt 1rteT Zita r 1e V 3n POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS 179 With a little manipulation this becomes p np n 1 aquot quot n 1n 2 aquot39 9 Just as in the previous example this recursion formula enables us to express a in terms of ac or a according as n is even or odd a2 a0 03 2 p 2ah 2 3 2 1 3 a4p 3 2pp 174 p am p 3p 4 p 1p 3p 2p 4 quot5 45 3 5 i quotquot p 4p 5 a6 56 a4 pp 2p 4p 1p 3p 5a 6 0 a p5p 6 7 67 5 p 1p 3p 5p 2p 4p 6a 7 and so on By inserting these coef cients into the assumed solution y Z axquot we obtain y a01 pp2 1x2 PP 2p4 1p 3x4 pp 2p 4p 1p 3p 5x6 6 a1E P 13517 2x3 P 1p 3gp 2p 4x5 P 1p 3p 5p 2p 4p 6x7 q 7 10 as our formal solution of 7 19 D39lF FERE HTI tL EeIis rIess when p is met an intege139 eseh series in hrseliet39s has rsdtius ef eerwereienee R 1 This is mesti essilyr seen by using the reeursiert ferms ls 939 for the first series this ferrnuls witth 1 replseed by 2n yielsds HE 2x En 1 HE 12 z w 2 y r i P 2 quot 1 irlxlisuixsz Zen 12rr 2 as H Lz 00 she simiilsrlgr fer the seeendi series As befere the feet that each series has peisitrise radius ef eeirwergenee justi es the epemttients we have pertermed snd shews thst 10 is a valid setutien of T fear every eheiee at the eenstsnts sat enri ei Eeeh hreeketetzt series is a psrtieuisr selutiien and since it is elesr that the f39IliIi39i IGiiii39I C39IiI391EA de ned by these series are Iiinesrilff inriep e39r1dentt 10 is the genera selutiein ef T39i en the iirntierssiir Is i 1 The fumetiens de ned thy rg ere eeiied Legendre f s ffi il and in gertersl they are net eiementisr Hewe39srer when p is 3 henriegstise integers true if the series terminates end is thus s peIynevrnisi zthe rst series if p is even end the seeenel series if p is edd whiie the ether ees net and remains an in nite series This ehstersstitenr leads its the psrticutsr EDiUii tTS Di T imewn as Legendrte peiynemieist whtsse prefperties and pZ1iit393 ii S we discuss in Chapter P We new sply the t39t teti 1G t39i ef these essmplies the estslsiish the feiiewieng gerreirsi theerem sheet the nature ei seiutiens near eir dinsrr points Tiheerem iI Let r be en ere riiery peim ef the 39d WErE 1rE eq ee t iern P E APEW essay 0 111 end Jet and is he erhirrery eenstems Then there existsquot er unique femeri39en y sji that is enefytie er J is e set39trti en emf equerien 1 bB in v eerreirt neighberheed sf this peirre sect setis tes the imquottiJei eemfifiens39 yl xi l as end quot39e 41 Ferrhermere if rhepewer series teseensiens sf Pr39 emf Q rje are ueiid en en int e39rvs 39 is Isl 3 R R 3 men the power senies eseensiern ref this sexttstien is aim veils en the sense im39ereeL Preej39 Per the sake ef eerwertienee we restriet eur i1rg L1tl7 II 1vEJt l39t its the ease in which rE 0 This permits us to werrrk with 39peiiwer series in J rsther than Jr r and irwelives rm reel KIUES ef ger11erslitjr With t39his slight simp39iii ea tieh the Lhypethiesis tlii the theerem is tthst 0 em are enelyrtie st the Uriggins tsnd thiereiere have pewe r series espansiens pi ms pas 12 I 1 R I end firm i ll i E21 quot 39 ll the s is I POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS pNn that converge on an interval x lt R for some k gt 0 Keeping in mind the speci ed initial conditions we try to nd a soiution for 11 in the form of a power series y2axquota0axa2x2 14 n0 with radius of convergence at least R Differentiation of 14 yields y Z n 1axquot a 39 2a2x 3a3x2 15 n0 and Q yquot 2 n 1n 2a2xquot n0 2a22 3a3x3394a4x2quot It now follows from the rule formultiplying power series that Pxy39 Z Opxquot2 n 1axquot n n0 5 pn kk 1gtaxquot 17 n 0 k 0 and Qxy qxquotiaxquot qkaxquot 18 On substituting 16 17 and 18 into 11 and adding the series term by term we obtain 2 n 1n 2a2 2 pkk 1ak 2 qkakxquot O kO k0 nO so we have the following recursion formula for the a n 1n 2a2 2 k 1pkak1 qak 19 k0 For n O 1 2 this formula becomes 20 poa qoao 2 3a3 pa Zpoaz qlao q0a 3 4a p2a 2pa2 3p0a3 qzao qa qoaz These formulas determine a2 a3 in terms of a0 and a1 so the resulting series 14 which formally satis es 11 and the given initial conditions is uniquely determined by these requirements 132 DIFFELREANTIAL ECiUATIU139H39S iSiupp0ase imw that we can pirimre that the i39 s 14 with its cinE icicnisi da niedi by formula 19 actuiailiy CU39 39ii rgES fft f ix i R Then by the geiinaraill ths ljf uf puwer series it will follow that the fcnrtmaii pBii39Eti n5 by which 14 was quotmade to satisfy 11J i1f Eii39mWiSE i iemntiaitii n imuitiipiicatinn5 aund termiibrytenn additiinna airei juati eid and the pram will be cnmplete This argument nut easjr X give the details in Appendix A wheire thvzy can b imitt39Ed c0niveiimieniihF W any readctr whIr wishes in dim 5 few nal remarks are in cznrdeiir j uur Exaimiplregs WE 3nca11ntercdi mniIJ5r what are 1mnwii as rwmrerm rEacumiian A for the cuune imzienis inf l unkimwni wriesi 5Iiiutimis The 5implicity of ithaae formuias makes it fairy easy to deitermine the ggeenerafl 0 of the resi1ij139ting series and mi mbiain prieciisie informatiin aib i1t 0 raii of E IIZ l1 i iuquot IEI gEaI1EEii H W39iEgt i P is aipparent fmm immuia 19 thia39t this simp1 licityr invest to be eixpeicitw in general in imust 63533 the best we czar do is in nd the radii f i2U 39VEe139gEi EE Bf tl1 e series iezipansiimnsi Di P and Qix and in mnciudle from the thgmeim that the radius fur the sieriiesi 5 riutin must he at ieast as large as the small r Df thesis numbei3 Thus for Liegeiinidir s requiatimi it is ClE 139 fmm B and the familiiar exanstinn 1 2 L V y 1 xri x x Rw1 1 it that r 1 for both Px and z jm iill We ihiemfoire iknnw at once witihwi further Cv lE39i1i ELtiI1 that any s0 iutinn of the farm y 0 ai x must be valid at least cuntheimewai1 lxl r 1 PRUBLEMS i Find the geinegrai 5miuii n of 1 xi yquot Zxy 231 E U in tearims nf p w f series in m Cani jmui ex4pres5 this S h1Ii 39by m EEi1Elt of E1i 1E i E E I39gquot fumtians 2 CU1i15idE31quot the EquatiUm yquot r 39 y U 3 POf its general suiuiin E nx in 11r1 e form 5 aQyix an yg xjn where yx agnd yi Vxi are p wwer sei39iiei5 a USE the mtin iesi to verify that the two scriea 39 x and Ayzfxj cmnverige fur ali I as Themrem A aaserrts amp ljcjli Show that 3Ih39 is the serics expiansi n emf a i 39 USE this facti to find a sgcond iindependanti soluiirzin by the metih d vi Setctitnn 16 and iuIw39incie jmur5eiif that this secmind scuuti0n is the functimi rrjj fnur1d in ar Verify that iihe equa39t i n yquot y xy D M a 391fEE 39tEI39I7139i reiciursiiinn forimuia and nd its iserics a iutiiinnis je i1I39 an d yx xj such that p J 319 3 K yiiii i D bi mm 2 K wmi S THEDTEim A guarantnauessi that both zserieqgs convcrgt far all x iNit3ti13 hmwi niiE ui1t this would be to priw E by w cwrkiing with the scries iihemsalveesi rowan SERIES SOLUTIONS AND SPECIAL FUNCTIONS 183 4 The equation y p 1x2y 0 where p is a constant certainly has a series solution of the form y E axquot a Show that the coefficients a are related by the threeterm recursion formula 1 n 1n 2a2 p a a2 0 2 b If the dependent variable is changed from y to w by means of y we quot2quot show that the equation is transformed into wquot xw pw 0 c Verify that the equation in b has a twoterm recursion formula and nd its general solution 5 Solutions of Airy s equation yquot xy O are called Airy functions and have applications to the theory of diffraction A a Apply the theorems of Section 24 to verify that every nontrivial Airy function has in nitely many positive zeros and at most one negative zero b Find the Airy functions in the form of power series and verify directly that these series converge for all x c Use the results of b to write down the general solution of yquot xy 0 without calculation 6 Chebyshev s equation is 1 x2yn xyr 8 0 where p is a constant a Find two linearly independent series solutions valid for x lt 1 b Show that if p n where n is an integer 20 then there is a polynomial solution of degree n When these are multiplied by suitable constants they are called the Chebyshev polynomials We shall return to this topic in the problems of Section 31 and in Appendix D 7 Hermite s equation is y Zxy Zpy 0 where p is a constant a Show that its general solution is yx a0yx a y2x where 2 2 3 y1x1 x2 x42PP 62quotP 4xe S 4 Sir George Biddell Airy 18011892 Astronomer Royal of England for many years was a hardworking systematic plodder whose sense of decorum almost deprived John Couch Adams of credit for discovering the planet Neptune As a boy Airy was notorious for his skill in designing peashooters but in spite of this promising start and some early work in the theory of light in connection with which he was the first to draw attention to the defect of vision known as astigmatism he developed into the excessively practical type of scientist who is obsessed by elaborate numerical computations and has little use for general scienti c ideas 1 HI EFF E R EquotN1TI u L EQUXTIGNS and k 2 1i 22 1 J lquot 15 41 I P Sana JSIS me ew eeee Ti By 39Tite erem A heth 5EI39tEi5 etitw erge fair till 11 Veri r thie tlirec tir In If p is a tiennegtaititre integequotr then tie not these eeries terminates and is thus 3 pDi339n eimiai yEr if p ie e veti and if p i5 tJtid whiIe the ether temeirie an in nite eetiee VE1if that f 1T pi 1 1 E 3 5 quotthese Eeiyttntniele are 1 xi 1 as 2Jt 1 x339 1 41 tf t Jrquot l It is tzieer that the eniy peiynemiel eeiutients ef Hennitequote equatiett are etnsteint multiples of the pe1yttIemaiie E165 E l iiIJEd in bi Timee eenetant rimitiplee with the preperty that the terms eenteiningt the itigliest pewetrs ef x are ef the term Tint are denoted 3 HJ ant eeiited the tHermi te pe39iyrtnmitti5 Veritfy that H xs 1 H391t i H2ti39 433 2 Hgfxj 8x3 121quot Ht 431 112 and tH5gtt B 1 x3 item 42 Vteirifiy that the pelynemiae iieted in C are given by the genem1 fermtuia V H5 12 E H 2 1 K U th Jam In F LZJ pE tE1iI39E B we shttw39 thew the fermuie in d een be dediieetl ftem the series in at we 39ZJl39393lquott eeterel et the imeet tieetiult ptepettiee ef the Hiermite pei3rnemiteie and we shew il 39rie y hewi these piynemtineie eriee in a fun dementei prehiem ef qtiantum meehen iest REGULAR SINGULAR We reeeil that e peint x is a sfigtiier peint of the di etetttiai eqiuetient WPamp QM M it me er the ether er both ef the eeet eient funetiene PJtS and Qr fails te be analytic at x In this eeee the theorem and methede ef the premiums seetierl tie net epapiy and new itlteee are nteeeeeary if we wish te stttdy the eetlutiene inf P neat x This is El matter ef tI39 l3II391SiEiEt l39quot5fiE ptaetieal impetritenee fer miatny dti etrenititel eiqiuetiett5 tthet arise in phyeieei prebiem5 have singttiet peinte and the eheiee ref pityreieeflly atpptrepritete eeluttiens is eft en detetminzed by their Kb39ElII quotiquot39i I39 near these points Thus Wi39liE we Iquotligi39l39I want tee aveid the E iItgI tiEIEquot peinte ef 3 dti erentiel equatimt it is preeiseely theee peititrs that usuaiiy dtemend particular ettentient As a simple egtamplte the erigin is eieeriy e eitiguiet peint of N n 3 y at 7 D 9 1 rowan SERIES SOLUTIONS AND SPECIAL FUNCTIONS 185 It is easy to verify that y x and y2 x 2 are independent solutions for x gt 0 so y cx c2x 2 is the general solution on this interval If we happen to be interested only in solutions that are bounded near the origin then it is evident from this general solution that these are obtained by putting C2 0 In general there is very little that can be said about the solutions of 1 near the singular point x0 Fortunately however in most of the applications the singular points are rather weak in the sense that the coefficient functions are only mildly nonanalytic and simple modi ca tions of our previous methods yield satisfactory solutions These are the regular singular points which are de ned as follows A singular point xo of equation 1 is said to be regular if the functions x x0Px and x xo 39Qx are analytic and irregular otherwise5 Roughly speaking this means that the singularity in Px cannot be worse than 1 x x0 and that in Qx cannot be worse than 1 x x02 If we consider Legendre sequation 287 in the form 2x 1 pp 0 y 1 xzy 1 x2 y it is clear that x 1 and x 1 are singular points The rst is regular because 1 1 x 1Px and x 12Qx quot W x1 x1 are analytic at x 1 and the second is also regular for similar reasons As another example we mention Bessel s equation of order p where p is a nonnegative constant xzyquot xy x2 v2y 0 2 If this is written in the form 1 x2P gt yquot39i y it is apparent that the origin is a regular singular point because xPx 1 and x2Qx x2 p2 are analytic at x O In the remainder of this chapter we will often use 5This terminology follows a time honored tradition in mathematics according to which situations that elude simple analysis are dismissed by such pejorative terms as improper inadmissible degenerate irregular and so on 39E UATfi39 JH395 esse1i s eqestient as an iiilusitrrative iexsmp Ie ends in CL tepteir its vii i s and tih ir appiieettitenst will be essrnineti in eettsitierebie deteiil News let us try te undeszrstantd reesenst bei39iieti the dei tniititien 0 e s reguiszr sitngulsrr peint Te simptif39y imstters we ems sssume P singutiet paint 39 ieeated at the erigin fer it is int then we sen iaiws339s menses it te thee teritgint ehengtiI39tg the imtepentdenett vsrisblie ftwmt it te r E x Out starting peintt its feet that the general term ef as futnettien malytie at x t39 ii is as six nix e V Ii As at eensequenejett the erigim wii eerteinify es singular paint eff 1 if Px p L ha 4 bi bgxz s sends n39 cs QLE C cg ii six t teas 7 1 and st iesst ene the cee ieients with negative subscripts is netneeire The type of sveiutmn we are siming at fer 1 for rressens tilgat wiii appears ibeiew is e i qussi pewrer seties ef the 06 pO E Im g eels itgig e g n p pn 2xnt3 t J Pq where the iespnent V mm be e negstiveV mtegerz s frsetietni er sen an irra39tiet1si real number We will see in Prebleimst 6 and Tr that twee iinaiejpendent seiutiens i this ltindi are pessitbiet enitjr if the ebe39vet espressiens ffti P and Qx de nest eentaint tresipeetivelyr mere than the rst term er mere th t1 the first twe 39tierms te the ieft ef the ee ttsttsnt terms bf and em An equissltent sitetement is that xPx and x3Qr must e snteiytite et the erigin smdi seeertdieng te the e n1itiet1 this is pteeiseiy what is mtiesnt by staying that the sit1guisr peint ix 2 U is regular The nest qeest39ien we st39tte mpt the answer where d we get the idea iht t series Vg the term 3 pi be suitable seitutiiens fer equstigem 1 near the reegetiitsri siniguis r pint J U Zj tinis stitage the semis steeemd erder linear equetien we can SDW13 eempletei near a singt11iar point is the Euler equetien diiseessed in Ptrefbteim 1751 xi PIE st E 0 iii If Ub is writtltet39i in the term P Q tfiyse cs 0Z se Pix E p is sn d i tix qfxjj then it is eieer that the erigin is s regelsr singular peintit whetnever the teeintsttsets tR and q are net beth eerie The seiutitens eat this equtsttitent presiitde s vent 5tlgE SJ1iiquot uquotvE bridge te the genersi ease se we brie y recall the deteifls The key to IildiI1g tihes e POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS solutions is the fact that changing the independent variable from x to z logx transforms 4 into an equation with constant coefficients To carry out this process we assume that x gt 0 so that z is a real variable and write and 1 J l f XquotZ xzdz xdz dz dxx 22 x 2 When these expressions are inserted in 4 the transformed equation is clearly dzy dy 39P139ampqy0 6 whose auxiliary equation is m2p 1mq0 7 If the roots of 7 are m and mg then we know that 6 has the following independent solutions 0112 e39quotquot and e if 7712 ml quot and ze 39Z if m2 m Since e x the corresponding pairs of solutions for 4 are quot12 x39quot39 and x if m2 5 ml 8 xquotquot and x quot39 logx if m2 ml If we seek solutions valid on the interval x lt 0 we have only to change the variable to t ex and solve the resulting equation for t gt 0 We have presented this discussion of Euler s equation and its solutions for two reasons First we point out that the most general differential equation with a regular singular point at the origin is simply equation 5 with the constant numerators p and q replaced by power series 2 Z 3 yr P0 Pix xP2x y 510 Clix xzqzx y 2 0 rowan saunas SOLUTIONS AND SPECIAL FUNCTIONS 189 and y a0mm 1xquotquot392 a1m 1mx quotquot a2m2m1xquot To nd the coef cients in 13 we proceed in essentially the same way as in the case of an ordinary point with the signi cant difference that now we must also nd the appropriate value or values of the exponent m When the three series above are inserted in 12 and the common factor xquot 2 is canceled the result is a0mm 1 am 1mx a2m 2m 1x2 1 0 xa0m 1 am 1x a2m 2x2 1 1 2 a0axa2x 0 By inspection we combine corresponding powers of x and equate the coef cient of each power of x to zero This yields the following system of equations 1 1 a0mm 1 2m O am 1m m 1 aom 0 14 a2m 2m 1 m 2 0 am 1 0 As we explained above it is understood that an 0 It therefore follows from the rst of these equations that 0 15 1 1 mm 1 mE This is called the indicial equation of the differential equation 11 Its roots are m1 and m2 and these are the only possible values for the exponent m in 13 For each of these values of m we now use the remaining equations of 14 to D IF39FEE ErfI L E LmTI r3Hs E l ul t an 13 in tarms of am Fm a 1 we obtain r E3 V 0 W SE Z Hr 1 1 12 5 2 2 We therefm e Vhmre the fIEqwing Ewen Fmbenius series snlu139mns in each nf which we hmre put 1E j 1 4 y1 S J 1 Ex 163 I o I6 y x39 21 A x 333 P E 150 THESE scrlutiig ns are dearly iI39ld77E pE EI 1t far I 3 0 SD Illa genertai sn1u tiI3n nf M cm this interval is p P c1x1 51 Ex c3 I g I 2fx1 The prnblhaam Bf detAesnniming the inteWal mf rI11wErg JVncze fur the two pnwer series in parentheses will be di 3 I1SSEd in than 11am 5ecAtiDnV was l k zclumly 31 the way in which l arises I2 is E S in same that the irndicial equati n Of the mere general di eVrentiaI Bquatiwzm 9 is mm1m4qmq a mm POWER SERIES soumons AND SPECIAL FUNCTIONS 191 In our example the indicial equation had two distinct real roots leading to the two independent series solutions 16 and 17 It is natural to expect such a result whenever the indicial equation 18 has distinct real roots ml and quot12 This turns out to be true ifthe difference between m and m2 is not an integer If however this difference is an integer then it often but not always happens that one of the two expected series solutions does not exist In this case it is necessary just as in the case m m2quot to nd a second independent solution by other methods In the next section we investigate these difficulties in greater detail PROBLEMS 1 For each of the following di erential equations locate and classify its singular points on the xaxis a x3x 1y 2x 1y39 3xy 0 b x2x2 12y x1 xy 2y O c xzyquot lt2 xy39 0 d 3x 1xyquot x 1 2y 0 2 Determine the nature of the point x 0 for each of the following equations a y sin xy 0 d x3yquot sin xy 0 b xyquot sin xy 0 e x y sin xy 0 c xzy sin xy 0 3 Find the indicial equation and its roots for each of the following differential equations a x3yquot cos2x 1y39 2xy 0 p E b 4x2y 2x 5xy39 3x2 2y 0 4 For each of the following equations verify that the origin is a regular singular point and calculate two independent Frobenius series solutions a 4xyquot 2y y O c Zxy x 1y39 3y O 399 2xyquot 3 xy y 0 d i x2yquot xy x 1y 0 5 When p 0 Bessel s equation 2 becomes xzy xy xzy 0 Show that its indicial equation has only oneroot and use the method of this section to deduce that an 0 2n Y 22rxn2x I is the corresponding Frobenius series solution see Problem 26 7b 6 Consider the differential equation I I 1 39 y 3 gr 0 a Show that x 0 is an irregular singular point b Use the fact that y x is a solution to nd a second independent solution y2 by the method of Section 16 mFFEaEaTmL E LIM1wJH5 c Shaw that the secmmd swxnilutinn yg fmmd in b cagrmm be expressed as a FrDbe ni39u5 EEII39EiE C 5ird f the di 7rcntia Erquatin n if r i r r E y Ity In 0 wh re p and are nanzcm real numfber5 andC h and E are p15E39tiiVE in39tegr5 It is clear tThat J U is an irregular singuiiar fp i I if 112 1 Dr 1 2 3 If b 2 and c i 3 Shaw that there is nly ne pn5si39lJe vaIue of m for Awhich there axial a Frrbeniu5 sgiries s l39utiUn rjliV Shvcsw similarly that m satis es a quadrati erquat innaand hence wt can hope arr two F rubeniu5 serie5 suA1uVtim1s Carre5p an 39ing tn the rnmt5 of thies equat39inn if a39nI cmly if b 0 1 and E 1 bserve that these are exavtljr time mnditinns that c h aracterizI S D as an w akquot39 CW rgular singu laur quoti nt as quotp p E In 3 5IErnang Dr irregular angular pvm39nL E The dii ampreAn4tial qrnJatiLinrn yquot Ex E m 0 has E U as an irregular 5 ingu1a13939 puin14 If 3 is inse1rtied mm this equartVE4uunA mow that I and the a r139espunding F39r nbenius sreries quotquot sniutim1quot is the puwer SE39I39isE5f mi y pR nix nn l wfhic7h En uerge5 Aum gr at 0 This dem1mr 5at4rates that ver1 when a TFr h eniu5 5er iE f 39nnaL39hr satis es rssmch an equminn it is not naaEss5arily a valid 5nluIVimtL INUEDJ ur in the pmvicwus 5e tiDm was mainly dimctaz at mDt139Wat i H and terhniquci AWE mow E 39 fIE339Tlt the thVe07retical side Eif the p1wsblem f snMng the ermral semnd wider HI1E T iampq39uatimn J39 Pxy QIv 390 1 near the IEgUl f 5inguiar39 int x H The ideas dtvElDjp ed EIbmrB suggegst that we attempt E formal nalmllVatinn if any 5nli utim 1s of 1 that hawrve the Fr be niu5 fnrm y x 1n I mg agggtzi v 2 Wham Hujl U and x is 73 Anumber In b dj etE4m1in4eVd Omquot hvupe is that any f1mAal sznmtinn than arises in this way can be l gitimi gzd by a prUm and cstab1ished as a valid s lutimn Tm gnenerality of this appmach MM ai sn 5e1W 10 ilh1miwnaV39te the cirAcuVmstances under which cqmttirzmn 1 has cu y POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS 193 one solution of the form 27 For reasons already explained we con ne our attention to the interval x gt O The behavior of solutions on the interval x lt O can be studied by changing the variable to t x and solving the resulting equation for t gt 0 Our hypothesis is that xPx and x2Qx are analytic at x 0 and therefore have power series expansions xPx E pnxquot and x2Qx i qnxquot 3 nO n0 which are valid on an interval x lt R for some R gt 0 Just as in the example of the previous section we must nd the possible values of m in 2 and then for each acceptable m we must calculate the correspond ing coe icients a0 a1 a2 If we write 2 in the form 3 y x quot 2 axquot Z ax 39quot n0 n0 then differentiation yields yl Z anquotl nxmn l n0 and yquot Z am nm n 1x quotquot 2 nO x quotquot2 Z am nm n 1xquot nquot0 The terms Pxy and Qxy in 1 can now be written as 2 K M xquot quot2i xquoti am nxquot nO nl am nx 39quotquotquot n k w xm 2 2 n0 x quot 2 i pnkakm kxquot E pn kakm i k P0anm quot xquot k0 7When we say that 1 has only one solution of the form 2 we mean that a second independent solution of this form does not exist 13994 DIFFE39REH TIAL EQMTIUHS and 0T d r qxn2 m El l WnV39 lquotl xm qnmxquot 39 quot39 39 1 0 kt quot H xm139H T A when these expressions Em yquot P xy and Q y are inserted in 1 and the c mm n t39armr x is cancele d than thc di ferential eqmati un bccnmes x Jam M M 1 m np q3 fjL 0v p EF kI ksD and equating In zero the me ic iVenJ1t of x yixeldls the follnwing I39EEl1rSiU IIi fnnmlim for the an a m njt am n 1 Mpg 1 q H1 Eh sc m MPH 4 1 E 0 Pc Um wriming this aurrt Em 11e 3uccessive valuxss of ma get Wm p pE llp z a mpl 2 D u239m 2m 1 m 2sV D 1 q 3mp3 z gmi 1 q1 39 Q m nJm n 4 np q j mmp c Hl m H 1p gm 0 If we put fwfm mp qm than tfhesg equlatiuns become nf39m U HFrwmPE 0 E Us POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS 195 02f quot 2 o quotP2 42 a1 quot 1P1 Iii 0 afm n allmp q alm n 1pl ql 0 Since all 5 O we conclude from the rst of these equations that f m 0 or equivalently that This is the indicial equation and its roots ml and ml which are possible values for m in our assumed solution 2 are called the exponents of the differential equation 1 at the regular singular point x 0 The following equations give al in terms of all a2 in terms of all and al and so on The a are therefore determined in terms of all for each choice of m unless f m n O for some positive integer n in which case the process breaks off Thus if ml ml n for some integer n 2 1 the choice m ml gives a formal solution but in general m ml does not since f mg n f ml 0 If ml quot3912 we also obtain only one formal solution In all other cases where ml and ml are real numbers this procedure yields two independent formal solutions It is possible of course for ml and ml to be conjugate complex numbers but we do not discuss this case because an adequate treatment would lead us too far into complex analysis Thespeci c difficulty here is that if the m s are allowed to be complex then the a will also be complex and we do not assume that the reader is familiar with power series having complex coefficients These ideas are formulated more precisely in the following theorem Theorem A Assume that x 0 is a regular singular point of the differential equation 1 and that the power series expansions 3 of xPx and x2Qx are valid on an interval lx lt R with R gt 0 Let the indicial equation 5 have real roots ml and ml with ml S ml Then equation 1 has at least one solution 50 yl rm 2 axquot all 5 0 6 n0 on the interval 0 lt x lt R where the a are determined in terms of all by the recursion formula 4 with m replaced by ml and the series 2 axquot converges for x lt R Furthermore if ml ml is not zero or a positive integer then equation 1 has a second independent solution X yl xquot 2 2 ax all 0 7 n0 196 MFFEAEHTIAL Et39tJ t IquotEUtl5 an the same l7F1I39ErU f 39where in t39quottIquot 39I the an FE39 dEEE rt f E f in fEF LE f an by Kj 7m39rnttamp39tt PF mtquottquottaIFt m replaced by ml and agaifn the Sn rf i E r1t 39 E UEl39g39 5 fatr tr 139 In View at wthatt we haw aalmadly dmna the pit Izmf Df this theurem can be c mplEted 0 Shntwintg that in each 355 the amiss K aux m ntvettgEs cm the itttterval tn fr Readers when are interetsttied in the dataills at this argument quotwill nd them in Appendizt We EI39I39IP h 5ifEE that in a Speci c prhlemt it is muvh simpler tn substiytute the gettcrtatl rubanius series 2 ditrectly ittmt t tthe di i ierential mutatfmrnt than ttu use the rBctuIsti3tn arrt1ula 4 tan catlcutaxttte the rne 1 tiettts This T Et39El1SiiElTl39l fmmula nds its main aptpltilcatixtjn in the dEEicatut tE nquottquotEv139gE CrE pmciaf given in Appei dix 39Threurem A unttDtttutmatlyt fails t ntswert the qLie5tin Hf how to t nd a stecnntd s lutimn when tht dtiJ eren e mi h is E f or H Lptmsittitve itnttegert In UfdEr tn C quote39E1739 an itdFEa f the p ssitbilitttists here we disti3ng1tistht ttthree catctt CASE p If m1 312 thteret Ifa rIi39It exist H sttttxnd Fmbentius serit3 stJ utiDnt The rvther two CESES in bath of which 0 is Xw PDi fi h39E integer wilt be easier to grasp if wt il39IEEHZ ms 2 mg in the EiE ukrVEik frarmuia 4 antilj write y as u H w 0 x 5t 3t 39 ant l PnE H E v P w gilt E3 As We 1 U W3 the rdi asCul39tjr39 in talculating the an arises because fmt2 rt U far P0r teI39taitn pnsitive integeor mt o next txw cases dieal with this prmbteizm m P g i thc right side of SJ is not EBFO when H H HillEn there is nu pnssaiibm way of ttrnttinuing the CaIm 1la ti n at that me cricnts and there Ea n t exist a SEEDnd F t t benttu5 series sntL1tIinnt CASE d c tthte rigth39t 5173 Gt 3 happens tn be zuwrn wilten ftmg n a than an is U IE SftrirEtECi and be assitgnted any valuet Wh IE uBr In patti ru art we can put tI V0 and mntinue to tomputt thug cemcients withmmt HTIquot fUfthEf dtit tcul titt5tt AHEHCE in this case there d ess exist a second Fmtaeniust tsratrites L1iJC39l39l l The pra bJEm5 b l w wilt d matntstttate that Each of thESE thme pnts t sitsiiittittes attually l l rsl POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS p The following calculations enable us to discover what form the second solution takes when m quot12 is zero or a positive integer We begin by de ning a positive integer k by k ml mg 1 The indicial equation 5 can be written as m m1m m2 m2 m1 m2m mm2 0 so equating the coe icients of m yields po 1 m mg or m2 1 39 po ml and we have k 2m pg By using the method of Section 16 we can nd a second solution yz from the known solution y x quotIa0 alx by writing y2 vyl where 1 Ur WeIPx dx 391 3I ltpoxmgtctx x2 quot a0 alx 2 1 8 Po 081 Pix x2 quot a0 a1x 2 1 1 i u u pu39c gt xkaO alx 2e xkgx The function gx de ned by the last equality is clearly analytic at x 0 with g0 1a so in some interval about the origin we have b0 i blx bzxz 39 39 39 bo 0 It follows that A U b0x k b1X k1 bk1x1 bk quot39 SO b0x k1 bx k2 v k 117 j5b1logxbkx p9 and b x k 1 y2 ylv y1OIc 1 bklogx l bkx box k1 T bk1y1logx x quotIa0 ax k 1 If we factor x quot out of the series last written use ml k 1 7712 and multiply the two remaining power series then we obtain y2 bk1y1logx xquot 2 Z cxquot 10 n0 as our second solution 193 etFFEeEeTIaLt iEGUATij N F ermu1e 1 has only htnited value as he practice tnrml but it does yield severe greihe heft in fm39metien First if the expten etente email and are eqeetl then 1 and Abk1 i been P U en in this eese whieh is Case T ehere the term eenteining leg 3 de niteIy present in the eeeend seluetien Hewte verj if rm 4 em 1 is e sietiee inhtegeere then semetinrlee 3km ee 0 end the ltgerithmie terrn ie preeentt Case E ent1 semetimes hamp1 et J and theree is me tegeritthmeie tlj eee C The praetizfel di ieeutlttv here is athet we cennet readily nd tht1 beeeuee we have new direet means hf ealeutletting the ee e ciente in r litleny event we at least knewquot that in Cseeee end E1 ewhen bkE ell and the metheti f Frehetniue s enty partly euseeessfuls the general farm ef e eeeexndg eetutien 1 J I Ex Em be 1 11 where thee ten are eerteiitrt Mnknsewn eenetente that teen be dgettermtinted ye 5ehetitutiLng 11 di1 C39 jvquot ingfte 0 39iHEI39E tTiEHl equetientt Notice that this expereeetitent is similar te formula 29L1 but eemewhet mere etempIieet edt The equetietnt yquot E 3xyquot Met 4y 0 has ently tme Erelzzehitue sezrites eehltien Find it The etqeateiem 4e1yquot39 E 8ltr39239 432 1 has D Elly rjme Frebeniue series eehttien Find the gerterel EMlL1 Eii39lClI1 3 Fine twe iendependen39t Freheniue eerie5 eetuttiene eat eaeh ef the fell eeritng equetriene T ta mi r 35 at IJtIt 0 64 L E Jr We E 0 tcba 0 t at x my U Eesstelhs EqtJB39iEt ef eedter pt 1 is 1quot Jay I 111 10 SIhew that my re and that the eq39uetim hes enly ene i39Fre39hentiue series evelutinm Them it B ee5eE e et1etien ef etrder p H e y xy 2 1y 0 Slmw that re H13 L hut ithart ee39vertheteee the rerqeetiten hes twe iI1dtE ZIEt d t em Frehtentue series etutietae Thtetn nd 39thtem POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS p O 6 The only Frobenius series solution of Bessel s equation of order p 0 is given in Problem 295 By taking this as yr and substituting formula 11 into the differential equation obtain the second independent solution 1nFl 1 1 II y2ylogx 39 l22nn2 1 dH x2 31 GAUSS S HYPERGEOMETRIC EQUATION This famous differential equation is x1 xy c 4 a b 1xy aby 0 1 where a b and c are constants The coef cients of 1 may look rather strange but we shall nd that they are perfectly adapted to the use of its solutions in a wide variety of situations The best way to understand this is to solve the equation for ourselves and see what happens We have cab1x ab x1 x a Q M so x O and x 1 are the only singular points on the xaxis Also xPx quot 1 1quotc ab1x1xx2 cc ab1x Px and abx x2Qx1x abx1 x x2 abx abx2 so x 0 and similarly x 1 is a regular singular point These expansions show that po c and qo 0 so the indicial equation is mm 1mc0 or mm 1 clO and the exponents are m O and m2 1 c If 1 c is not a positive integer that is if c is not zero or a negative integer then Theorem 30A guarantees that 1 has a solution of the form yx Zaxquota0a1xa2x2 2 nO where a0 is a nonzero constant On the substituting this into 1 and equating to zero the coefficient of xquot we obtain the following recursion formula for the an a nb rt aquot1 n 1c n aquot39 3 l DIEERJVEMHAAL E U L39TI HE We Imw seft g 1 and B lE5l al the ntI1 r an in 5U39l3CES5i ILZ a4b V E aa 1brb 1 EE E 12ccm H ai g 1 1r a 0V 9 2 A 23c c 1 4 2 quotWith E E E Eie39 F3tS 0 bEE1D39139 E5 A V W ga 1b b 1T1 539 1 1 cl 1 2m I f z W E 1 I V2 T3c 1u 2 aa 1 v E f a H i lb bM 1 E e l fr 1 1Z M ncc 1 P n 11 Tg a as a This is knuwn as the hyergeamerrEc er e1 39 and is dekn o1ed quotby ma Symbol FA abEx It is callampd itihis 39becausB gencralizcs the familiar geumetVrici series as f 39 39D39Wl jZf when H 1 1 and E E b we obtain 9 E T IE 39 quot quot T 39 F 1 r If 15 01 b is EEFD GI 51 negatiVve inTeger the s rxi s 4i breaks off amzl is 3 polyn mial therwi5e t1391 3 ratiZn test shows ltharlz it mnquotver1ges for lxli c 1 since 3 givas u 391i39 1 quot f f f 393 J in Ta 1 E M as H m an 1 1 39 H I I I This mnvergm1cue hehavinr maid alga have ibcen prr Ed7ictrd frm the Vfaut that the singuIar PIlilIf clDsEst Im the UIfquotiil39TI is 1 Ar rdingy w han cf is not zero arr a nEga1ivB in7I egEr Fabm is an analrthic functimmFa calli d hypergmmeIric fur1Vcfi annun 39thE iVnteWal ig Aii 5 2 is the sirnplest particular 5nIL1t ivun 4f thc h3rpEreUmetric equatim1 hyper ge0meIric fll cti W has a gr1rat many pmperti es 0 whichw the must uhviou5 is P D P unaFlLer6d when car and b art intarahanged F17b x FbaVcx3 If 1 c is nm ZEl39 U1 3 negattive inteVger which IIquotlB3 cr is izmt a positive intaeAgerr thEn T39hmr em EUEA alsD tEII5 us thavl them is H second independentt S luti D1 1 nmr x s D with Exp0n3nt mg 1 3 This BAA 51Jmmary f iiJIquotna Er 39i39l5 I II39IEEIquot prnpermic s mm f39 u239 f urm in An Erd ilr39i Ed554 Higher TrarL539c nd nru39f Funcrinn v39 zrAJL L pp 5 a119g Mt raweHill New furky F POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS solution can be found directly by substituting y xquot a0 alx azxz into 1 and calculating the coefficients It is more instructive however to change the dependent variable in 1 from y to 2 by writing C y x1quot 2 When the necessary computations are performed students should do this work themselves equation 1 becomes x1 xzquot 2 c a c 1 b c 1 1xz a c1b c1zO5 which is the hypergeometric equation with the constants a b and c replaced by a c 1 b c 1 and 2 c We already know that 5 has the power series solution 39 zFac1b c12 cx near the origin so our desired second solution is y x quotFa 39 c1b c12cx Accordingly when c is not an integer we have 2 cFabcx c2x quotFa c 1 b c 1 2 cx 6 as the general solution of the hypergeometric equation near the singular point x 0 In general the above solution is only valid near the origin We now solve 1 near the singular point x 1 The simplest procedure is to obtain this solution from the one already found by introducing a new independent variable t 1 x This makes x 1 correspond to t O and transforms 1 into t1tyquotabc1 ab1ty aby0 where the primes signify derivatives with respect to t Since this is a hypergeometric equation its general solution near 1 0 can be written down at once from 6 by replacing x by t and c by a b c 1 and when t is replaced by 1 x we see that the general solution of 1 near x 1 is ycFaba b c11 x c21 x quotquotquot Fc b c a c a b 11 x 7 In this case it is necessary to assume that c 1 b is not an integer Formulas6 and 7 show that the adaptability of the constants in equation 1 makes it possible to express the general solution of this equation near each of its singular points in terms of the single function F POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS 203 in uence since many of the principal special functions of higher analysis are also related to it PROBLEMS 1 Verify each of the following by examining the series expansions of the functions on the left sides a 1 39 x Fpbb x b log 1 x xF112 x 1 c sinquotx xF2 gm 1 3 d tanquotx xF2 1 E x2 It is also true that e e lim F aba 35 b gt o b 3 2 f sinx x li1 r i Faa2 1 x2 g COSX Faa 2 Satisfy yourself of the validity of these statements without attempting to justify the limit processes involved 2 Find the general solution of each of the following differential equations near the indicated singular point 3 a x1 xyquot 0l 2xy 2y O x 0 b 2x2 2xyquot 1 5xy y 0 x O c x2 1yquot5x 4y 4y O x 1 d x2x6y 53xy39yO x3 3 In Problem 286 we discussed Chebyshev s equation Ux n p 0 where p is a nonnegative constant Transform it into a hypergeometric equation by replacing x by t 1 x and show that its general solution near x 1 is 11 x 1 xquot2 1 131x yCJ 2 2 F quotp 2 9A brief account of Gauss and his scienti c work is given in Appendix EUUAquotI1UN5 4 Cneider the dit erentiet reque39tien x P P Elxllye mt 03 w here is a curn5ten39tf 3 If p is not an integer nd the ge39nerel s0I1utquotien nteer x II in teirme ef hr39pe r39gE DIt1etriC fun etitenst E Wrrittte the general enlutient fmmd in e in terms of etemtenttary functions e when p E 1 the d39i errenJtieTl equetiten becemee E J391 i e yquot 1 3x Vyquot EL and the eIuti nn in be is me mnger the general tsel utien Firtdf the general 50luti n in this eeeet by the em etthAedi of Sietet nn P 5t1me tdi ierenttitel equet39im1e are ef the hrpetrgeurmetri e type even theug39h may net appear te be V tFind the generrel 5 Dlt11Zquoti at 1 V equoty z ety D nteer tlhe singular paint 1 J by changing the indeependentt varitaire ta I z e 3 0 that ebwtx i e ta L e Le T h E applyitng the di Ferenteiattiu tn ftmttuta in 3 tn the resutt eff Prebt emt 3 shew that the eel S E3hlt7lEiit IquotIS ef Chebyshtewe equetien Wh SE tdterivetivee e 7 1 1 39 are hemmded nearxt r 1 ere yt eF I Ce39nefNude that me enly epetlym mtial selutttiene esf 39Chtet bty5hevt t5 equatien are constant muultiptltes T 1 t ef Fn En P I The 39CJiE by5 hEUt petfyetemt m of degree H fie dented by Ttr famed de ned by V e r 1 11 t t Fn mg 2 quot nemeiatle tea the theery Bf apprmtime tEien dieeueeed in Appendix D where n is a nunnegative in39teger quotAquot Art eitnteresttitntg applieetit n ef theset pelya 32 THE POINT AT 1 tttTY It is eitem la si Ev iETbR j in th and pure tmathemeteicst ID study the 5nIutt ne DE Jquot PI y39A Q rt39 t quot3 U for large velutes of the inQIependeI1t verziatbtet Fm zintetteneet the quot39tquot I39i IblEi is ettirne we may want to knew ht1w the phjyeieal Sy t t l described by D Btu y 5 in the tdi tant f ill E UE when tratn5ien39t a1ieturbeneee theme faded z M39hThe ntetatjen Tjx Pi us39ed fbeeatuee Cheby5he39v s name was fDl39 39lEfW trtens39lAitereted es Teheb395refhtetr TeheL39weheff39 er T5ehehyeehetff x39dt x2 quot at 3 and i d3 2f2 d 2 y dxdx dt dxsdx tdtz 2tdt t 4 When these expressions are inserted in 1 and primes are used to denote derivatives with respect to t then 1 becomes n We say that equation 1 has x 00 as an ordinary point a regular singular point with exponents ml and quot12 or an irregular singular point if the point t O has the corresponding character for the transformed equation 5 As a simple illustration consider the Euler equation 5 4 2 quot39 0 6 Y x xzy A comparison of 6 with 5 shows that the transformed equation is 2 2 yquot y gy 0 7 It is clear that t 0 is a regular singular point for 7 with indicial equation mm 1 2m2O and exponents m 2 and m2 1 This means that 6 has x 00 as a regular singular point with exponents 2 and 1 Our mainexample is the hypergeometric equation x1 xy c at b 1xy aby O 8 uL EnunTmns Win nilnrendgyr knnnw that 8 has Lwn nite resgular Singznnilnr pints Jr 0 with nxpnnnnts U and 1 5 and x 1 with expnnennts and if n b Tn determine the nature of the p int x a an we snnb stitute p and 4 rirnctnly intsn 3 After an lintnt le rearrangement we nd that ting trnnsa finrmnd equation is n 11 an bn 2 c n A I n k n r1 I F1 0 E nqnnntinn has I S D as 1 regular sninngnnlar with indi cin1 nqnag nn O 1 1 n Mn ab U nr m njn m b p This shnwns that thn exponents nf Equation 9 cal I U are An and b so nqnnntinnn fl has Jr n n nm as 3 regular singular point with napnnnn t5 n and We cnnnlunln that the hypergnnmntrin eqnatinnn E has pnrnnacisnlgrn thrne rneguiiarn Si g39Ui l39 pn inrts 1 and with EI3Tl E Squotn ding enptnnennts D and 1 C n n and E i an H b and Ann and b In Apnndin E we djamprnnn3trntn that the fnrrn 01 the hypnnrgnnmntrinc nqnaatinn is cnznlnntnnly dntnrminnnd by the speci Enntinn f these three regular sninngnlnr nints tnngnthetr with the addnd reqninrnenment that at jeast nirle enxpnnnnnt must he niernn at nanh nnf the points 1 E D and 1 p Annthnr cnIa55ical dierxenxtialn equation nt cnnsitmrnblei innfmpsnrtance is the cnnnn nnnt hypnerg nmerric EqM I39i myquot 39C xjry my 1 1 0 Tn nnndInrstnn dn where this mnntnninn nnrnen frm and why it bears this nnmen wn quotECl39ILSiELBlT the nrdinnnnrny h3FpETgE mEr39ITiEnEL1lJ I i I1 3 in the form quot1 an s c n b 13E 5 my 0 11 If Ithn inndnpnndnnt varia k changed frnm 5 In I V b then nwen hnave 0 0 0 nnd 11 b i x1 5 vquot nn ta p i 39 y my 5 0 my bk 0 x b T where primfnns dnnennnte nrerinvatinvnes with rnnrspnnt tn 8 EqnnantLin e has regular nsinJgular pnints at x U x 2 Ab and Jr p it frnm 11 in that thn singnnlnr paint 2 2 is znnw mnbile If we int b 3 W than P2 Vbctcnnnlns 1CIn Thn singular pninrc at b has nvnidnn tn13r cnnlnscnd witnh the 207 POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS one at 00 and this con uence of two regular singular points at 00 is easily seen to produce an irregular singular point there Problem 3 PROBLEMS 1 2 Use 3 and 4 to determine the nature of the point x 00 for a Legendre s equation1 x2yquot Zxy pp 1y 0 b Bessel s equation xzyquot xy x2 p2y 0 Show that the change of dependent variable de ned by y tquotw transforms equation 9 into the hypergeometric equation t1tw 1ab a1ac1tw39 V a1acw0 If a and b are not equal and do not differ by an integer conclude that the hypergeometric equation 8 has the following independent solutions for large values of x 1 yFa1ac1ab1 x x and 1 1 hFFJb q1b a Verify that the con uent hypergeometric equation 10 has x 00 as an irregular singular point Verify that the con uent hypergeometric equation 10 has x O as a regular singular point with exponents 0 and 1 c If c is not zero or a negative integer show that the Frobenius series solution corresponding to the exponent 0 is Z 12 n1 aa1an1 H nmcnn 1f The function de ned by this series is known as the con uent hypergeometric function and is often denoted by the symbol F a cx Laguerrequots equation is xy 1 xy py 0 where p is a constant Use Problem 4 to show that the only solutions bounded near the origin are constant multiples of F p1x and also that these solutions are polynomials if p is a nonnegative integer The functions Edmond Laguerre 18341886 was a professor at the College de France in Paris and worked primarily in geometry and the theory of equations He was one of the rst to point out that a reasonable distance function metric can be imposed on the coordinate plane of analytic geometry in more than one way r 1 POWER saunas soumons AND spgcuu FUNCTIONS 209 converges and for this we need information about the behavior of the ratio bb as n gt 00 We acquire this information at follows Replac ing n in 4 rst by n 1 and then by n 2 yields M rn l n 1 nn b1 kz 1bk1 bkrk Mbr 0 and 2 M quot n 1nb 2 Z k 1bk bkrquot Mbr k0 By multiplying the rst of these equations by r and using the second we obtain M Z2lk Y 1bkl bkquotk rn 2k0 rMnb b Mbr2 n 1nb Mbr rMnb b Mbr2 1 rMn Mr2b mn 1b so b n T 1n rMn Mrz b rnn 1 This tells us that b 1xnlr bxrl Ix39 agt ii bxquot b 39 The series 5 therefore converges for x lt r so by the inequality a S b and the comparison test the series 2 also converges for x lt r Since r was an arbitrary positive number smaller than R we conclude that 2 converges for x lt R and the proof is complete Proof of Theorem 30A conclusion The argument is similar to that just given for Theorem 28A but is sufficiently different in its details to merit separate consideration We assume that the series xPx Z pnxquot 39 and x Qx Z qnxquot 6 H n0 n0 converge for x lt R R gt 0 The indicial equation is quotm 1 quotW0 q 0 M and we consider only the case in which 7 has two real roots m and mg with mg lt m The series whose convergence behavior we must examine IS 2 axquoti 8 210 DIFFEREHT L t L E UIAT1Q NS where an is an rbi tI39 f39 nnna m wntstant and tilts t39thfI an area de ned recursitvetly in terms nf an by f 1 3 T T K K Jan g F V q k W JU Uur task is EC prm39e that nth seirias J cinnvaxrgcs far 115 m m ml andii alga if 0y p and mi 0Z tits mlAt tat p 0sitive itmt ger 39 We begin by nmbserving that f can he tw rittten in n form ftm E m 0D E ml mggm 3 little cal ctL1latit nt thiss etnabl e5 us tau write H Ht 0 and T k 0 B mu af d Eo tsequentttly mm H1 3 MI E lmt mm W and Ii mz 0 HM PO 0 mat Let r be a pJ39U3itiv tntumber aucah tihat r 2 R Since the series 6 vEUn tPErgE tr 1 r there E1 ii539lS at cumstant 3 U with the pri partty that fp il F E i and q l rquot 0 M 12 fur all Hi If we jpuct p mg in 9 and tuse 10 aLnd 12 we ilirtain 3911 T 39 i imt tram we 0w T fit k quot H We now de ne a seqjutantce E 31 writing bu 3 man fair 039 5 H 5 E t5391r2 and 0 b t nan mm a mEbn 5 M e titmll k x 1 13 5quot fgrr H 3 ml t mgt It is EiE T that 0 E l N E for evEry FL We shall tpr rve that that sseries M cmtnvErge5 Fm ml at r and In achtEw3 this we SEEK at mnvenitetntt mcprtessti tn Em tTh Lrgntii bE139 39m libfmtl t1EptItaci11g H thy n t in t1t3 multiplying by an and using 13 tat simplify th result we btaim Ft 4 130quot 439 1 11W m2bni rttfriii rm E mgljbt H 1 POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS 211 S0 bi1 quot391 quotquot imi quot mzi Mim1l 391 1 b rn 1n 1 m m2 This tells us that bn1xnH bnl ixl lxl gt bxquot b so 14 converges for x lt r It now follows from 0 s a 5 b that 8 also converges for x lt r and since r was taken to be an arbitrary positive number smaller than R we conclude that 8 converges for x lt R If m is everywhere replaced by m2 and 11 is used instead of 10 then the same calculations prove that in this case the series 8 also converges for x lt R assuming of course that m m2 is not a positive integer so that the series 8 is well de ned APPENDIX B HERMITE POLYNOMIALS AND QUANTUM MECHANICS The most important single application of the Hermite polynomials is to the theory of the linear harmonic oscillator in quantum mechanics A differential equation that arises in this theory and is closely related to Hennite s equation Problem 287 is E2p1x2w 1 where p is a constant For reasons discussed at the end of this appendix physicists are interested only in solutions of 1 that approach zero as x gt 00 If we try to solve 1 directly by power series we get a threeterm recursion formula for the coefficients and this is too incon venient to merit further consideration To simplify the problem we introduce a new dependent varible y by means of w ye 2 2 2 This transforms 1 into dzy dy quotampquotx3 2xE2P 0 3 which is Hermite s equation The desired solutions of 1 therefore correspond to the solutions of 3 that grow in magnitude as x gt 00 less rapidly than equot2 2 and we shall see that these are essentially the Hermite polynomials Physicists motivate the transformation 2 by the following ingenious argument When x is large the constant 2p 1 in equation 1 is 0W EI IFFER39EHTAJL euwrrxeee neg1igi4ble ee mpa1red1 with IE ee r1 is epperexeimaeteelye dime A E FEE E W Aiia392 z is net tee eauterageeeus 1 guess that the mmeteieene w e eeletiense eef P eq e1iem We new eh eer ve that might be wquot re quot 2quot1 arm W e E i ee i VZ e and eiineee for large 1 the eseeend term atquot w can be negIeete d ee mpeired with the 0 P appears 39l l 1a1t W ee and w aequotquotE are itndeed apprmeimeate eeelutienequot ef 14 The rst ef theee is neltw dieeaerdeadi because it does net eappreaeh zer as lex 1W It is therefere rea semaVhe le to suppose that the exact eelutien of T has the ferem 2 1 where we hope that W funetien x has a sieemmeer etrue39tmquote than vW ri W heteeveer e e elhinke ef p reaSOning it weeks Fer we have seen in Problem 284 that Herermeitee eqeuateeieen 3 has a eweteeerm IEEllrsi fmimeula HWEE Neew 3 H L r T end else ethet this fermeula Ageneraetes twee eindependienet series eeleuteins 23 I M e eJ f2mp Mp M 39 2Te 4 5 W me1 we and r E z gt 0 y e x m3 g T 6 that eenverge fer aw y We nnw eemperee the rates ef gerewth of the functions ylix and e 393 Our purpeee IS Ate prrzwe that p d p d A w r p t as LI u em Exz if and e11ljr q the series for yn7r Abreaks eff i1IE is e pelynemeieale that lil al igf e11de11Iy if the paraem eetere has one ef thee mines 2e4 0 o o part P clear by l Hesepietel e rule To preeevee thee quot nljr part we assume that 0 Z and shew that in this ease the abewe quetient dlees not eap39preaeeh eemJ do p we use the feet that yli x has me firm quot1I m 2xEquotquot with its e eee I Ieientes determined by g and the eemlietien em L and also thawt em has thee series eepenerien e 0 b r3 where rowan SERIES SOLUTIONS AND SPECIAL FUNCTIONS 213 12 1zms so y1x a0a2x2a4x4 a2x2quot e 2 2 b0b2x2b4x b2x quot39 Formula 4 tells us that all coefficients in the numerator with suf ciently large subscripts have the same sign so without loss of generality these coef cients may be assumed to be positive To prove that our quotient does not approach zero as lxl gt 00 it therefore suf ces to show that 12 gt 22 if n is large enough To establish this we begin by observing that b2n2 1 a2n2 2P 2quot H bzn 2n Z d a2 Zn 12nlt 2 a 0 a2n2a2n 2P 2quot2quot 1 gt2 This implies that a2n2 3 02 b22 2 172 for all sufficiently large n s If N is any one of these n s then repeated application of this inequality shows that a2N2k gt k y gt 1 2 b2N2k b2N for all suf ciently large k s so a2b2 gt 1 or 12 gt 172 if n is large enough The above argument proves that y1xe quot2 2 gt 0 as x gt 00 if and only if the parameter p has one of the values 024 Similar reasoning yields the same conclusion for y2xequot 2 2 with p 135 so the desired solutions of Hermite s equation are constant multiples of the Hermite polynomials H0x Hx H2x de ned in Problem 287 The generating function and Rodrigues formula We have seen how the Hermite polynomials arise and we now turn to a consideration of their most useful properties The signi cance of these properties will become clear at the end of this appendix These polynomials are often de ned by means of the following power series expansion 00 tn e2xtI2 2 H 7 0 fl t H0x Hxt Equot19 Q DFFE1ElENTiI uL EQLI39 gTIUH395 The fu tiDH ef is amed the gwener iing funcrimz of the Herm n polyrrmmiam Th39is dB i tiDI1 has the audvantagc Bf seV crienBy Em dedunzzing pirVpertie5 wf iihe x and the nbvirnus weaknessv of being tnmiiliy unmtivated We shalVil therefore derive W Enimm the SEIAEES snl41mtin4ns 5 6 All Vpmlyn mi al 5 lutirJns Elf 3 are 0bIaVimed fmm these seri5 Es by replacing p an integer r E 0 and niultiplyiing by an ariftrar39 mnstant Tlmy all have t11 e form 39 an I V V a3 x quot agx quot 2 ax ml A amt r gx HI a5w r wherc sum Pg Vwritten ends with an or mix acmrd in as H is veve n at mid and Vits coe cientsA are relatad by A mn m A W jk 1jk 2 3 t shall nd gg an in terms nf um and tn this and we replace k in 3 by 0 a and gsv21 g k z E 0l T T an2 k 1N t 3 or E 2 E kk 1 ii H E I 2 8 Lettiilg k be 1 r 5 2 1 4 am yilelds E 0 0 i n H jF1 D E 3L 2 E4 4 3 5 pj 23 2 P 3 4 T 4n Sjr nn 0S7 2rI 3rH 4 rz 23 C 4 2 6 I in j POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS 215 and so on so n nn 1n nn 1n 2 3 n hxax 2 2x 2 2224 x 4 nn 1n 2n 3n 4n 5 n6 39quot 23 2 4 6 x L quot39 n 1n 2k 1 1 kn n 2k 539 2k242k x This expression can be written in the form n lquot 2 n hquotx aquot oquot1k22kn 2k 2k where n 2 is the standard notation fo the greatest integer sn2 To get the nth Hermite polynomial Hx we put a 2quot and obtain n In2 Hnx k 01quot 2Xquot392quot 9 This choice for the value of a is purely a matter of convenience it has the effect of simplifying the formulas expressing the various properties of the Hermite polynomials In order to make the transition from 9 to 7 we digress brie y The de ning formula for the product of two power series Oantquot 0 butquot E Y akbnktn n0 kO is awkward to use when the rst series contains only even powers of t 10 at2quotn20 butquot What we want to do here is gather together the nth powers of t from all possible products a t2quotbti so 2k j n and the terms we consider are akt2quotb2ktquot392quot The restrictions are k 2 0 and n 2k 2 0 so 0 S k S n2 and for each n gt 0 we see that k varies from O to the greatest integer sn 2 This yields the product formula co 00 00 n2 2 at2quot2 bntquot 2 2 akb2ktquot 10 nO n0 n0 kO 216 D39FFEi ENTML Eaumxcms If we now insert 9 itmtn thgec right siampd e of T and U15 IE1 WE bt in e a 0 m g Pw 2 P d En s In i Wu p2b 0g 1444 5 11 N1 E um tan h M which establishes T As an appliatian e T we Anise R0drEgu 39 fmirnm39a fur the Hermite p lyI1 rni4ia4is 39 1393 E d 1 x 4e ma In View 39f fmrmuia 269 fr the C E 1iiE tE Elf a wax s eVries T i ldS V V nftrim E awe f 393 39 A 6 W rr3 ln j m 4 39e Ilil J D39 we intmducE 3 HEW var iable 3 08l 1 I an 1156 the fact that 31 T Ba39 0 k tihen Siifllr li E U m1rVre5pnnls In 2 rj the exprltessimn last wiitten ljneciamges an an 11 nexl 1 E r1 i TA w m 13 T FE LIE 1 and the pr f is C mpl t rtVhumgnnaViityi we krmw that for each n nnnagmive intager n the lmcti n rU rU 12 caMed the Herrnire fumrrianz of rdifr H1 apprnazchas zeraraw as M E T and is a 5 mlIL1t inn of the 4diffEmntial equatinn K HK 1 r3 w 1 L113 An impmtam pmperty mf thhe5e functi ns is the fact that I I dx I e iHmxHxdx 0 K H at 0 F 143 This I EIE11 i rI is often expre ed by 5ayirIg Airhaft the HE Imi tE fTLmcti Uns are rrhoganm on the imewal m my rowan SERIES soumons AND spscuu FUNCTIONS 217 To prove 14 we begin by writing down the equation satis ed by wmx w 2m 1 x2w O 15 Now multiplying 13 by w and 15 by w and subtracting we obtain d Exw39w ww 2n mww 0 If we integrate this equation from 00 to 00 and use the fact that w39w w39w vanishes at both limits we see that 2n mJ ww dx 0 which implies 14 We will also need to know that the value of the integral in 14 when m n is I equot 2Hx2 dx 2quotn7t 16 To establish this we use Rodrigues formula 11 and integrate e 2 dx fm ex HxgtHltxdx lt 1gtquotfm Hxgt by parts with u Hx du H x dx dn x2 dn l x2 dv dxquot 6 dx 1 dxne Since uv is the product of equot and a polynomial it vanishes at both limits and co 2 oo dn 2 f es Hxgt12 dx lt 1gtquot Hltxgt dx es dx n2 O0 dn 2 2 1 J H xdxn2e dx 12quotf Hfquotxequot 2dx Now the term containing the highest power of x in Hx is 2quotxquot so 5 DIFFEREHquot TML EElU LTDH5 Hjifiixi 2 2 and Was itntcgrai ia I i E dx I wi ix 2 wiiich is the desired resuiiiitfz These l39thDgiD iitjf prape 1fti es can be used to sezxtpiamdz an binary fU IEti TiJ I in a Hermite seriasiz m L an mis M tap If we f f id fnrmiallyi the cne ients at can The fmmci by muittippiyiiing 17 by gHx and 39 i1aitir1g tetrmi by term tfmm 09 m By 14 and 16 this gives 395 H W equotiiHm xfxi V an g E quotquot1Hm xiHr 2 am2t miVj it rn D am 30 r eipiiaIi11tg p by M a1Je4mumnamp uamp Tut i P This f rmai prwcedum suggesits ha mathematical pmbiem at determinin c nditinns mm the functimt fxi that guarantee tfhaitt W is vltalid when the gs are E tied PmTbElEm5 emf this kind are part at the genraLi thE fquot at Fth gitZl EtiI fuincttininsi S rm diitrevzt pihysicai ppiiE 391E39iiiDrlS Eff orthiDgnial expants39innsi like it are discusser in Appendtiic s A and B of Chapiter The hannnnit usizIIatmr As we sitaitedi at tjhe beginning the mtatthuematiii cal ideas dIesvie iitpampdlt abmre have ih 39ir mgaizn apptiiitraIioIi in qrtlantuirri meuzhaznitts iadequate diiscu5s39i Dn at the un deriytiing pi11r5icai cmncqatsi is cilearijy theyand the scape mf appentdix NiIE V39BthEiESS i5 quite easy tn understand the role played Ebvyp the Ehfcirimitie p iiiyiin miaits H5 Jr and the wrElt5p0ndiitn4t Hermitt funcitiuni5 e quotquot iH I In Seictiian iwe ainiiaiiiyzed the Ei Sii i i h fm iiv unisi1latn Ii whicht can he thought at as a particle rat 111355 cnnstirairied to mew along the x axis and band tn the Eq1i1i1 i39IIt itl tI1 DSiitii n I 71 by a treatsirintg tfarce kxi Equatti n inf mmqinn P dgx m dig it h Thu f t2quot that thin irmtrgral mi ii fr nt D in 0z i5 Dftm pr39 r5 39d in ElEmentrar3quot caicLJIuis SEE Pr biEm POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS and with suitable initial conditions we found that its solution is the harmonic oscillation k x x0 cos y t m where x0 is the amplitude We also recall that the period T is given by T 2Jrmk and since the vibrational frequency v is the reciprocal of the period we have k 4r2mv2 Furthermore since the kinetic energy is mdxdt2 and the potential energy is kxz an easy calculation shows that the total energy of the system is E kx a constant This total energy may clearly take any positive value whatever In quantum mechanics the Schrodinger wave equation for the harmonic oscillator described above is d2 l 8n2m 1 1 7 E quotquot2quot 1 lt19 where E is again the total energy his Planck s constant and satisfactory solutions 11x are known as Schrodinger wave functions If we use the equation k 4Jr2m39v2 to eliminate the force constant k then 19 can be written in the form 8n392m dxz hz The physically admissible or civilized solutions of this equation are those satisfying the conditions 15 2Jr2mv2x21p 0 20 12 9 0 as x gt 00 and I ltplz dx 1 2 These solutions the Schrodinger wave functions are also called the eigenfunctions of the problem and we shall see that they exist only when E has certain special values called eigenvalues If we change the independent variable to u 1 2n V1nx 22 3 Erwin Schrodinger 18871961 was an Austrian theoretical physicist who shared the 1933 Nobel Prize with Dirac His scienti c work can be appreciated only by experts but he was a man of broad cultural interests and was a brilliant and lucid writer in the tradition of Poincare He liked to write pregnant little books on big themes What Is Life Science and Humanism Nature and the Greeks Cambridge University Press New York 1944 1952 1954 respectively 220 IZHFFEREHTIAL EeUsTmsss then heeemes digs V213 A k A e end eenditiesms 211 heseeme 2 as 0 es lei es and P iyixfli due En P Eh 24 Eseept fer mquotQE 3Iti 39 Eqll ti 23 has esse y the fesm evf equshien 1 sh we knew that it has seluftinns sstissfying the rst JI3 di EiIZl hf 2 3lh and 39I39 yquot39 if EENW 52 1 r E wn 5 25 fer seemes neinsnegst39isse integer n We also knew that in tfhis esse these S h1ti elf 23 hhssses the farm 1 where e s se enstsn1t we new iirhpese the seems eehd tisen of 241 end use JJEL then it fhililews that 1 W C 21 nsJ The eigenssfeu39neti esn E IquotfES di fg 10 the eigezwslue is thsereeferes dstvm 39ilMv V 0 0 EampM3Hn iis 25 WhETtquot 22 gives u in terms hf e F39hysieists hsve 3 deep pmfessisenses interest in the dsetsilesd pres perties ef these eigesnfiunesions For us hewesvers the prseli ems is eniy sh illustrstien es the eeeurrenee ef the Hlerjmite polynomials so we will net pursue the msttser snjy fusrthsesr hse3rend quotP iIHTifI39Is hut that ferm39uls 25 yiheilds the 539QquotquotC 3L EslCI q1 r1fi2 EQ energy Ieueis f the hhsrmehic DEEi m t El1 This means that the senergyr nr may assume enly these discrete quotv usE39 S whsieh hf CDILIFSE is very dif fEil39 EF l sfresm the eshrssespensdisng ehsssierels siteussiinsn desse139ihesds sfbseve The simpElersI ejnerete spphestien hf these ideas is to the vih139stiens Amatien ef the sstems in s d39i Is 39m iE molecule When 1his phehemehen is studiseui eape1imkeshtslhhl y the ehsservesl E11quotEampI giEs are feuhd st he preeisealy in ss eeerd with Ka a pd HERMITE Charles T ermiite 1S272I9 IVs sIfmIE ef the most eminem French mstshemsstiesisms ef she ninet eenth een tmsr3si was pssrsise uisssrly disstsihguis hesl fer the ehsgsne es end fhigh sstisssie qusity ef his week As s s7tudfen1 he eurted dissrssrterr hy negiee1sihng his 139UL llil1E ssssigned week the sturhvquot the eEsssEe masters ef POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS mathematics and though he nearly failed his examinations he became a rstrate creative mathematician himself while still in his early twenties In 1870 he was appointed to a professorship at the Sorbonne where he trained a whole generation of well known French mathematicians including Picard Borel and Poincar The unusual character of his mind is suggested by the following remark of Poincar Talk with M Hermite He never evokes a concrete image yet you soon perceive that the most abstract entities are to him like living creatures He disliked geometry but was strongly attracted to number theory and analysis and his favorite subject was elliptic functions where these two elds touch in many remarkable ways The reader may be aware that Abel had proved many years before that the general polynomial equation of the fth degree cannot be solved by functions involving only rational operations and root extractions One of Hermite s most surprising achievements in 1858 was to show that this equation can be solved by elliptic functions His 1873 proof of the transcendence of e was another high point of his career Several of his purely mathematical discoveries had unexpected applications many years later to mathematical physics For example the Hermitian forms and matrices he invented in connection with certain problems of number theory turned out to be crucial for Heisenberg s 1925 formulation of quantum mechan ics and we have seen that Hermite polynomials and Hermite functions are useful in solving Schr6dinger s wave equation The reason is not clear but it seems to be true that mathematicians do some of their most valuable practical work when thinking about problems that appear to have nothing whatever to do with physical reality APPENDIX C GAUSS Carl Friedrich Gauss 17771855 was the greatest of all mathematicians and perhaps the most richly gifted genius of whom there is any record This gigantic gure towering at the beginning of the nineteenth century separates the modern era in mathematics from all that went before His visionary insight and originality the extraordinary range and depth of his achievements his repeated demonstrations of almost superhuman power and tenacity all these qualities combined in a single individual present an enigma as baf ing to us as it was to his contemporaries Gauss was born in the city of Brunswick in northern Germany His exceptional skill with numbers was clear at a very early age and in later life he joked that he knew how to count before he could talk It is said that Goethe wrote and directed little plays for a puppet theater when he was six and that Mozart composed his rst childish minuets when he was ve but Gauss corrected an error in his father s payroll accounts at the age of three His father was a gardener and bricklayer without either 14 See W Sartorius von Waltershausen Gauss zum Gedachtniss These personal recollections appeared in 1856 and a translation by Helen W Gauss the mathematician s greatgranddaughter was privately printed in Colorado Springs in 1966 DlFFERLE HTL I Eeuarteias the means er the inelinatien rte help develop the talents of his sen Fertunately hewever Gauss s remarkable abilities in ntental eentputialtien attracted the interest ef several in uentials men in the eemntunity and eventpualljy hreught him to the attentien ef the Dulce elf Brunswick The Duke quotwas impressred the and 1unclertnek te suppuvrt his further edueatienf first at the Careline Cellege in Brunswielat 192 ilquot95 and later at the lU1r1 iquottfEI ilI39 ef Gtittingen l795 1T98 Art the Cpareline Ciellegie Gauss eernpleted his masitery ef the classical languages and eztpleredl the werlcs nit Newten Euler and Lagrange Early in this peried perhaps at the age at fentrtleieni er fteen he disemrered the prime nurnher theerein whieh was i39inalily pressed in 183 after great e erts by many mathentatieians see eur netes en Chehysherv and Riemann He also invented the rnethed ef least squares fer rniniirniaing the errers inherent in ubseraatienal data and erneeiisedr the Gaussian er nermal law of distributien in the theory at prelrahilit n At the university Gauss was attlraeted by phileilnegy but repelled by the 1 Etil lBmEftiC51 eeurses and fer a tinte the direction f his future was uncertain Hewaeser at the age elf eighteen he made a wendexrful geentetrie disee39rery that eaused him he decide in fasveir ef mialthiematiies and gave great pleasure tea the end ef his life The ancient Cireeiks had lrnewnn mlerrandeenipass eunstruetiens for regular pelygens ef 3 4 5 and l5 sides an far all ethers ehtainablei frern these by hiseeting angles But this was all and there the matter rested fer II yearns until Gauss selved the pirehlemi enmipletreliy He proved that a regular pelrgen with 1 sides is enlns truetiihl ei if and enly if n is the preduet of a pewer ef and distinct printelntiin1 bers self the fferrn pk 22 1 In particular when 0i123 we see that each If the eeirrespending nurners pk 3 J5 J11 39 5 is prime an regular purlygiens with these nuntbers at sides are eunsatnruetihle I 15 During these years Gauss was almesit everwhelrned by the trrent ef ideas which heeded his mind He39hegan the brief ntes at his seieutli e diary in an eftert tie reeerd this diseetreries since there were far tee many to werlur out in detail at that time The first eintry datecl March Si 17 states the eenstrnetihility ef the regular plygen with 17 sides but even earlier than this he was peneatrating deeply inter several unesplere d cenxtxnents in the theery ef numbers In 1795 he diseesiered the law et quadratic re eipr eeiitr and as he later wrete Fer a whole quotyear this theerern terinfrented me and abserhed my grveateist eherts until at last I i5ietailst ref seme elf these eenstriuetiiens are given in H Tietee Fnereus Prrehlants sf il tatherneIi39rr ehap IX tfrragaleeh Press New YDrl I965 POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS found a proof At that time Gatnss was unaware that the theorem had already been imperfectly stated without proof by Euler and correctly stated with an incorrect proof by Legendre It is the core of the central part of his famous treatise Disquisitiones Arithmeticae which was published in 1801 although completed in 1798 Apart from a few fragmentary results of earlier mathematicians this great work was wholly original It is usually considered to mark the true beginning of modern number theory to which it is related in much the same way as Newton s Principia is to physics and astronomy In the introductory pages Gauss develops his method of congruences for the study of divisibility problems and gives the rst proof of the fundamental theorem of arithmetic also called the unique factorization theorem which asserts that every integer n gt 1 can be expressed uniquely as a product of primes The central part is devoted mainly to quadratic congruences forms and residues The last section presents his complete theory of the cyclotomic circledividing equation with its applications to the constructibility of regular polygons The entire work was a gargantuan feast of pure mathematics which his successors were able to digest only slowly and with difficulty In his Disquisitiones Gauss also created the modern rigorous approach to mathematics He had become thoroughly impatient with the loose writing and sloppy proofs of his predecessors and resolved that his own works would be beyond criticism in this respect As he wrote to a friend I mean the word proof not in the sense of the lawyers who set two half proofs equal to a whole one but in the sense of the mathematician where 12 proof O and it is demanded for proof that every doubt becomes impossible The Disquisitiones was composed in this spirit and in Gauss s mature style which is terse rigorous devoid of motivation and in many places so carefully polished that it is almost unintelligible In another letter he said You know that I write slowly This is chie y because I am never satis ed until I have said as much as possible in a few words and writing brie y takes far more time than writing at length One of the effects of this habit is that his publications concealed almost as much as they revealed for he worked very hard at removing every trace of the train of thought that led him to his discoveries Abel remarked He is like the fox who effaces his tracks in the sand with his tail Gauss replied to such criticisms by saying that no 16 See D W Smith A Source Book in Mathematics pp 112118 McGrawHill New York 1929 This selection includes a statement of the theorem and the fth of eight proofs that Gauss found over a period of many years There are probably more than 50 known today 17 There is a translation by Arthur A Clarke Yale University Press New Haven Conn 1966 DIFFEHEhlTtAIs taseusswtess selfrespeeting a1rel1iteet teases the seaelding in place after eempteting this hhiltdjsihgs NeserIshelesse the dih i eutsty esf reading his wearlss grteataty ehainderedt the dJihusien sf his Jidieas CraiLIss39s dleetmat dissetrtatietns 1 was anether lTti39lESl tne h the hhteray f mathematies After ssesera39l ab rtive attempts by earlier mathematteia1scl Alembtert Ewes Lagrangte eLapJtaee ttshe fuhdamsehs ta theeem sf algebras was here given its first satisfaetery pirveefg This theorem asserts the he isteneet ef a rest wr eemples met tart tan pelynemist equattien with teat ear eernptezs eeemetieentss Gauss ss success inaugurasted the age ef E3iSlTB tC preefs which ever sinee have p tayed an impm39tsam part pare matahsemaquottietss Fursthermere in this i fht gpretef he gave femur allstegether Gauss apaptearsst as the earliest mathemaatsieiah ta use eemspteas nrutmhezrs and the gE quotmEII quot ef the complex planes E eempatettse een denesee 1 The ntest perquoti zt of Gasuss ss life was heasity weighted teward appii etd mathematics and with a few eesteeptiens the great wealth ef ideas in his diar3 39 and netteheetks lay in sus pendied s39nimatfien In the last decades ef the Eighl f h cenastussrsy 39mahy SII Uquot I ELl395 were s earehing f t a new planet bEIW E39EIVEJ the Ihi t s ef Mars and J sphere where e dequots law at shggestteds that there naught ta he ene The rst ans ftssrgest ef the numerous amine ptanets lcnewn as asrtereids was dtseesezred in that regieh in so and was named Ceres diseewrerjrs imniesatlr eehteided whim an asteaishirmg puahtieatiee S the phiiliesetpther Hegel when jeered at astmnemers fer signerstiang ptthitltestetphsy this seikenee he saiid eeulld have sasestd them from wasting their e ertas by tdemems Ttrtating that as new planet eeuld pesstibty exist Hegel eehtinued his easretesr in a stirr1tita1 itsin ans later arese he even greater Lheieghtst set eh1msyquot ehtfus eatien U httertunat ely the tiny new planet was di teustt ta see under the hest ef eireumstanees sand it was ss ete39n test in the qH the sshy near the sun The sparse ehrservastiehat diets pesed the pmhtem eff EZ iC Ul 1i i sg the erhim with sef eient aeeuraey the leeattet Ceres again after it had moved away frees the sun The sstrten emers hf Eurtepe attempted1 this tlztash witheeut sueeess fer many measthss Finsaltyu Gauss was attraeted hy the ehsIllenge and with the aid eff his msetthe d ef llleast s q1Jsres and his unparalletesdss skill at numteiraiteal eesmpu tatien he detesmined the tesrshttiti rate the asstmtnemers where te leeks D their t lBSETtfpeEK and there was He had sueeeted ed in redis eeverh1g Ceres after all the experts had fsilseda 39HThe idea at this presquotft is seer eleartjy espliKa39in edlh3r px Keih Efs memsiry quotrt39azhsmsaIEes frem an Aduasszeed 5E H iquotEwfl f f pp lD11 D394 Devser New rquoterlLi 19459 1quot See the last few pages at De Orl itt s sIsanstammsf usL I P 0r wimems1 Hegel39s 57h7mH39ehss Wzsrke Fremh1ann s erlag Stuttgau39t 39 POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS WK This achievement brought him fame an increase in his pension from the Duke and in 1807 an appointment as professor of astronomy and rst director of the new observatory at Gottingen He carried out his duties with his customary thoroughness but as it turned out he disliked administrative chores committee meetings and all the tedious red tape involved in the business of being a professor He also had little enthusiasm for teaching which he regarded as a waste of his time and as essentially useless for different reasons for both talented and untalented students However when teaching was unavoidable he apparently did it superbly One of his students was the eminent algebraist Richard Dedekind for whom Gauss s lectures after the passage of 50 years remained unforgettable in memory as among the nest which I have ever heard 2 Gauss had many opportunities to leave Gottingen but he refused all offers and remained there for the rest of his life living quietly and simply traveling rarely and working with immense energy on a wide variety of problems in mathematics and its applications Apart from science and his family he had two wives and six children two of whom emigrated to America his main interests were history and world literature international politics and public nance He owned a large library of about 6000 volumes in many languages including Greek Latin English French Russian Danish and of course German His acuteness in handling his own nancial affairs is shown by the fact that although he started with virtually nothing he left an estate over a hundred times as great as his average annual income during the last half of his life In the rst two decades of the nineteenth century Gauss produced a steady stream of works on astronomical subjects of which the most important was the treatise Theoria Motus Corporum Coelestium 1809 This remained the bible of planetary astronomers for over a century Its methods for dealing with perturbations later led to the discovery of Neptune Gauss thought of astronomy as his profession and pure mathematics as his recreation and from time to time he published a few of the fruits of his private research His great work on the hyper geometric series 1812 belongs to this period This was a typical Gaussian effort packed with new ideas in analysis that have kept mathematicians busy ever since Around 1820 he was asked by the government of Hanover to 20 Dedekind s detailed recollections of this course are given in G Waldo Dunnington Carl Friedrich Gauss Titan of Science pp 259261 lIafner New York 1955 This book is useful mainly for its many quotations its bibliography of Gauss s publications and its list of the courses he offered but often did not teach from 1808 to 1854 hn m Fs eiaasrnaL seemsens supervise a geadetiie survey hf G kingdnn39n and asari ensi asrpeets f this taskineinhdhiiineg extensive eld week and meany thediens IhrianhjgnIlathiens eeeupied fern an nun1her efi yearns it is natural tn suppese that a mind like his wenlei heave been wasted en mesh an assignmeint him the great ideas of seienee are been many snange wiaysh These apgpareihnthijy unre wharding iaibevrs mquotesnltedi in me u his d eepes1I send mnst iaIsreeaehing eentrihutihnnsi in pure n1 ahthhein1atiese witheal which Ei39I1Sih ii if enera theery ef rseiiahtisity weuid have heen quite impessihiei Gaanssis geedeti e mark was eetneerned with the precise meiasuremenht eff large tnieahnegies en the eanthfs snrfaeeh preiviided the stihnanlnhs that led tn the eidieas hf paper D isqanisi innes gyenenli es eirea E HpE iES carves 1827 in Kwjiilis h he ieunded the inhitrineeseie di ierential gEi quot7lB IIi hf general eurve snirfaeeisfi In ilzhis werh he Ii II39UdU39C E d eursileinear fC I39di5iIllt 3quotEES H and 1 en a ssnriaiee he ehtained the fnndam entahi qiuhiamalzihe dTi 7erential inseam is R dfhi 2F du due KV en fee the elein1ent ef are length aha which zrnakes it pessihie te determineee geedesie enwes and he fernsiulhartenzi the eevneepts hf Gaussian enrvahture and iirntegral euhrvat39nrgteii39i N mahinh seeir e results were the faineus ahenereme egregiaim whiehn states 39thal the Gaussian sun39ahtur39e depends nnliy n D F7 and F and is tihierefasre inieahriant under hendihng and the Ganssehennets meerem an integral euhrraters fer the ease ni as geiedesieetrianhg1e which in A general pj is the eeantrial feet Bf rnedernh di erentiail genhrnetiry in the large Apart Emmi 4 detailed iseeseri es the crux ef Gsahussis insighht lies in the weird im rinsie in he hshnwed hear in stndy the geiemetry inf a srurfaee by nfperathing enly en the surface itself and paying ne a39ttentinn in the siurmundinhg space in which it lies Tn make this n1n139e enneret e let Us imagine an inte igenxt twn dirnensinnal ereahne when ihnhahits a IIf aEquot but has ne awareness esf a third dimensieen er ef anything net en the srnrfaeeg if this e139eatnr e is eapahflle sf rneving sheet niheasuring distances aien the surface and uieitermining the shnrtest paitshh geealesie from nne peint in anther then he is aisei eajpahle nf measuring the Gausaiain eurvatnre at any fp i ii and hi eriea nsge a rich geen1e39lry en snrrfaeea and G geensetry will he Euciieiean eet if and enly if v Gaussian enrsathnre is evrerjywhere eeirn Whhien these eeineeptieins are generalized the more than thwn eiiniensinns then they epen the deer the Rieirnannian gieemeetrF tenses analysis and the ideas hf Einstein Anee itehier great week of this period was his 1831 papeer en hiquadrahtie ii A rlzranslaiien hjg A Hii IeheiI II and J Mnreheael was published under the tite Genernquot In1IesIign39Iinns nf Chirereri Surfaces by me Raven Press Hea letti New Yn rt in 1935 2 These ideas are eaplai nen in nemeehniieai Iangnagie in AC Laneae rs harem i7irsIeir1 and She Cnsmie iirinr 39n 039rder ehani Interseiie1neeWiley New Ynrh elm5 POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS 227 residues Here he extended some of his early discoveries in number theory with the aid of a new method his purely algebraic approach to complex numbers He de ned these numbers as ordered pairs of real numbers with suitable de nitions for the algebraic operations and in so doing laid to rest the confusion that still surrounded the subject and prepared the way for the later algebra and geometry of ndimensional spaces But this was only incidental to his main purpose which was to broaden the ideas of number theory into the complex domain He de ned complex integers now called Gaussian integers as complex numbers a ib with a and b ordinary integers he introduced a new concept of prime numbers in which 3 remains prime but 5 1 2i1 2i does not and he proved the unique factorization theorem for these integers and primes The ideas of this paper inaug urated algebraic number theory which has grown steadily from that day to this From the 1830s on Gauss was increasingly occupied with physics and he enriched every branch of the subject he touched In the theory of surface tension he developed the fundamental idea of conservation of energy and solved the earliest problem in the calculus of variations involving a double integral with variable limits In optics he introduced the concept of the focal length of a system of lenses and invented the Gauss wideangle lens which is relatively free of chromatic aberration for telescope and camera objectives He virtually created the science of geomagnetism and in collaboration with his friend and colleague Wilhelm Weber he built and operated an ironfree magnetic observatory founded the Magnetic Union for collecting and publishing observations from many places in the world and invented the electromagnetic telegraph and the bi lar magnetometer There are many references to his work in James Clerk Maxwell s famous Treatise on Electricity and Magnetism 1873 In his preface Maxwell says that Gauss brought his powerful intellect to bear on the theory of magnetism and on the methods of observing it and he not only added greatly to our knowledge of the theory of attractions but reconstructed the whole of magnetic science as regards the instruments used the methods of observation and the calculation of results so that his memoirs on Terrestrial Magnetism may be taken as models of physical research by all those who are engaged in the measurement of any of the forces in nature In 1839 Gauss published his fundamental paper on the quotgeneral theory of inverse square forces which established potential theory as a coherent branch of 23 See E T Bell Gauss and the Early Development of Algebraic Numbers National Math Mag vol 18 pp 188204 219233 1944 4 eiFrEeEr39IriiL efusriies39ee miath en1atiiies5quotquot As usuiajl he had been 39ii1i iii7ig abeiit these matters fer Amen yeerfss and amevrigi his dieemrer39iriesi were the divergpeinee I39iiEFEm else celled Genesis Iheererrr eff medern veetier anaiyeie the basic rrieari ireiiie t11eer em fer iiermeniie iunetiensi the reiry pewerful statrernent which lieter became kinewn are 39 39fDiiriei1ieit s prineipie and was nally proved by Hilbert in irave d1ieeuessedi the pub39lis iied prtien of Genesis Viral achievei rnerrt but the urnpubigiishieidf and privetre part was iaflmeer eqiieiiy q pTeS5i eei Miuefhi ef P6u ieeme re iigfh39l ieinly after hie d eath when a great queriiitgy eff metetriai from his neteTbeeke and eeiienfti e eeirreepeiirider1ee was eerefLiirli3r anelyeieiidi and included in f eelieetei wierkei Hie scienti c diaryi bee gnelreadiy been meritirerledrg Thie little imeeiiieii of 19 pages erie ef the niieet precieue dCiC39Ui i1Ei1i5i the i1 ir5Iii 39lijr39 ef mritheimeiticsr uf ii w umii 1898i when it ewes rimmed iamreng famiriy papers the seeeeier1 ref ene ef Gaueee grendeencsi It exteride irein 1 re b and eensie te ref 146 very ermeiee stetem ernisi ef the results ef his irwee39iigetiene whrieih efien eceripzied him jjer weeks er rneiriths All of this meteriiel me1ee it aburiderriiy eieer that the ideee Gene eeneeived rend werked 139I tJ Y eeri derieibiiie detail but kept to iiimskeif weuldl have rriede him the gireetieet rriaiheimatieirerri ef his time L he had pi7Zii3 iiSiTZIE i ihem arid den ei niethirig eiee Fer exampiiei the iheery ef ifil Ei39i7Dii39iS ef a tompiee variable was ene ef tiquotJJB imejer EEEDiI39liPiiiEi1I BIliS ref nineteenth century methemiiaties arid the eeri39irel ffacts ref this dirseipiine are Cauerhg ei iritegreli iheererri and the iTeyier erirl L lif t erpeeeirens if an arrriehitiei iximctieri fi33 i In a ieiter w39ri irer1 In his ifrierid Bessel in 4 Gauss ezipiieiilry states Ceiiehyir theererr1 and then remerke This is a very beeutifui theerern wheee fairly 5irripie pref i will giirie en a euitafhfle eca5irL it ie C i fi i i wiiih emer be utifuii 39ii iiIiiiiE which are eeiieerne d with series exipeneiene quot3quot5quot Timer rneriy years in ediireneei ei Iiil i ei eeiieilhr rredIited with thieee imiperterit diieeimeriiesi he ianew C aui eihrr ei tiieerem arid prieheibily kniew bethi series ierpeneiiierie Hewever ifer some ireasen the H iiiiabil euceeeiew fer puhiiretieri did net arise p gpeseibie expienaytiein fer suggesitei by his eemmerite a Vl tt VEr te iweifgerig Beiyai e eiieee frienrl from his univ ersi ty years 0 iwihem meiiinteilned e iifeieing iEDi39lquotEEipU idE C ZI is net icniiewieidige Ema the act ref iliieerinirig net peeeeeeieri but the act of geitirg there iwihieh greriie the greeiecet 2 iGeerge Frrieen 5 i39quotEeea r en the iampipaplieatien ei Meti iemai ierii riellyeiev in iiie Theeriee ei Eieetriei ljr eml Pviegnietiierrrf Ii32iHjI was inegfleeierl end eilriieeti renripi eIelir Liniiirie39w39r1 lJi39iliTii it wee repriiri39Ieri in i di 15 See Gia39uee39e Werim wii i83quot5iquotquot ii i9 i ii eerie VIII p 91 iiiilliiiii POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS enjoyment When I have clari ed and exhausted a subject then I turn away from it in order to go into darkness again His was the temperament of an explorer who is reluctant to take the time to write an account of his last expedition when he could be starting another As it was Gauss wrote a great deal but to publish every fundamental discovery he made in a form satisfactory to himself would have required several long lifetimes Another prime example is nonEuclidean geometry which has been compared with the Copernican revolution in astronomy for its impact on the minds of civilized men From the time of Euclid to the boyhood of Gauss the postulates of Euclidean geometry were universally regarded as necessities of thought Yet there was a aw in the Euclidean structure that had long been a focus of attention the socalled parallel postulate stating that through a point not on a line there exists a single line parallel to the given line This postulate was thought not to be independent of the others and many had tried without success to prove it as a theorem We now know that Gauss joined in these efforts at the age of fifteen and he also failed But he failed with a difference for he soon came to the shattering conclusion which had escaped all his predecessors that the Euclidean form of geometry is not the only one possible He worked intermittently on these ideas for many years and by 1820 he was in full possession of the main theorems of nonEuclidean geometry the name is due to him27 But he did not reveal his conclusions and in 1829 and 1832 Lobachevsky and Johann Bolyai son of Wolfgang published their own independent work on the subject One reason for Gauss s silence in this case is quite simple The intellectual climate of the time in Germany was totally dominated by the philosophy of Kant and one of the basic tenets of his system was the idea that Euclidean geometry is the only possible way of thinking about space Gauss knew that this idea was totally false and that the Kantian system was a structure built on sand However he valued his privacy and quiet life and held his peace in order to avoid wasting his time on disputes with the philosophers In 1829 he wrote as follows to Bessel I shall probably not put my very extensive investiga tions on this subject the foundations of geometry into publishable form for a long time perhaps not in my lifetime for I dread the shrieks we would hear from the Boeotians if I were to express myself fully on this matter 28 a The same thing happened again in the theory of elliptic functions a 27 Everything he is known to have written about the foundations of geometry was published in his Werke vol VIII pp 159268 1900 23 Werke vol VIII p 200 The Boeotians were a dullwitted tribe of the ancient Greeks POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS 231 so cos n6 is the real part of the sum on the right Now the real terms in this sum are precisely those that contain even powers of i sin 6 and since sin2 6 1 cos2 6 it is apparent that cos n6 is a polynomial function of cos 6 We use this as the de nition of the nth Chebyshev polynomial Ix is that polynomial for which cos n6 1jcos 9 2 Since 7x is a polynomial it is de ned for all values of x However if x is restricted to lie in the interval 1 S x lt 1 and we write x cos 8 where 0 s 6 s 71 then 2 yields Tx cos n cos x 3 With the same restrictions we can obtain another curious expression for 1jx For on adding the two formulas cos n6 1 isin n6 cos 9 l isin 6quot we get 1 cos n6 icos 6 isin 6quot cos 9 isin 9quot 1 cos 6 V1 cos2 6quot cos 6 i1 cos 9quot 1 Qcos 6 Vcosz 6 1 cos 6 V cos2 6 1quot SO Tltxgt ltx ix2 1quot x x2 Ir 4 Another explicit expression for 7x can be found by using the binomial formula to write 1 as cosn6 isin n6 Z cosquot quot6i sin 9 m0 We have remarked that the real terms in this sum correspond to the even values of m that is to m 2k where k 012 n2 Since i sin 9 i sin 6 1quot1 cos2 6quot cos2 6 1quot h we ave W2 cos n6 2 cos 392quot 6cos2 9 1 k0 29 Thesymbol n2 is the standard notation for the greatest integer ltn2 DIFFERENquotI39TmL E uAn N5 and Ihe139efaare p p q 5 H Mng2ampcv K J 39 W Q5 mgx Us In 5 It is cilaar quotfrom 4 that x q 1 and Tr xi but fm highBr val uAe5 of n 39I x is must aasilyr ccrmpurced fromu a meursim1 j If we writic ms 319 crs 1 103 n s 9 csr1 DB sin 9 sin 1 Bg and L cos rm 2 f C05 6 H ME A ms 1205 1 5 1fB 1 sin Esin gr A B then it fall ws that c sn cos rs 29 S 305 1105 1 w If we use and I pl fi cs 9 by 12 than this trigDVnometrric tidLEntity gives the dE5ir ed recu39rsion fnrrmula X Z By starting with EVx 1 gajn T3x 3 we nd from 6 that TVx 42r 3 E E xja 7 X E 312 IrV g x2 1 anti an on The hy39pe rge melrie flzmm To Establish a EIDI l EEti n betweEm Champibyshev s dii39 arEntial Eua am and the C3hsel3yshev39 p ly39n0miaL5 as was have just de ned Vthemi we use the fact that the pmiEynn 4mial y Tx Abacnm s the functia n y ms 15 when the 39variabEer is changed fmm x to AHA Ab means uf P F 2 ms E New F fun ctim1 y msn is clea iy a 3Dluti0n Df Elm d iffeVr39enArciaE eq uVatinm I and an easy calcuIati0 n sxhmws that clthanging the wlriiablre fmm 9 Vback In tran5iImnns T im Chebjyshev 5 Equa1 inVnA V V V T 1 E x1 h 1 may 0 We thereVf are knmw that y E 3 Lpmyn miial snlmimn of 3 Em Fmbtem Vft6 teilg 115 that the tummy poh n um4iaI salutinn5 If B have the I and Si EE 4 implies that 139TJ1 1 far every F 11 1 quot39V 1 and cF M H E W If we cnclmde tVhat F 2 0 1 1 v F m n39 3 39 x form EF 1 H 9 rowan ssnuas SOLUTIONS AND SPECIAL FUNCTIONS 233 Orthogonality One of the most important properties of the functions y6 cos n6 for different values of n is their orthogonality on the interval 0 S 9 5 Jr that is the fact that I yy d6 cosm6 cos n6d6 0 ifm n 10 0 0 To prove this we write down the differential equations satis ed by y cos me and y cos n6 yj m2y 0 9 and y n2y 0 On multiplying the rst of these equations by y and the second by y and subtracting we obtain d Eyv y ym m2 n2ymyn 0 and 10 follows at once by integrating each term of this equation from O to Jr since y39 and y both vanish at the endpoints and m2 n2 0 When the variable in 10 is changed from 9 to x cos 9 10 becomes 1 Tmx7Lx 1 V1 xi This fact is usually expressed by saying that the Chebyshev polynomials are orthogonal on the interval 1 S x S 1 with respect to the weight function 1 x2quotquot2 When m n in 11 we have dxO ifma n 11 1 2 i 7 Tx for n i 0 dx 2 12 3 2 quot1 1 x Jr for n 0 These additional statements follow from at quot f 0 J c0s2n9d6 2 Or 0 Jr for n O which are easy to establish by direct integration Just as in the case of the Hermite polynomials discussed in Appendix B the orthogonality properties 1 and 12 can be used to expand an arbitrary function f x in a Chebyshev series fx E aTxgt lt13 n P E U LJTl HE The same fmrmafl prcadura aa ba39fare yaieldaaa tzha aaamcients 14 and an F E a ax1fx p M 15 1 1 E 762 39 5395 far at Andi again atha trues mathematical iaaua ia the pm lwlaama of mdiaing ca11ditiaas under which the series 13 wim the an da nacl by 14 and 1S actually canaargaa ta fx Pb Ininimaa pmugparty The Chebyshear fprTblam we aw asiadar is ta asa haw closely the function 6 can be aaapjparaximatad on the intanral 1 5 x E 1 p l ra0 miaEa a1c ajxa an nfdegra aa H 1 that is In see haw small th Ililamhar T 39 n39f1x I E a quot 39H17 an 15 73521 can be p by an aappmpmiata afhaiaa 0f the cana 39iaaianta P in tam is equivalent to the farll 0awing pmblaama aman all palynomiaala Paxf 1 aa a Exquot h i aux a0 of dagriaza 1 with leading vzaemIient 1 ID minimize the numbaar tmax PVxf 5 15x51 M and J paaisaibala ta w a p ILquotl39I i 39li I that atataainsa tlnia minimum avalas It quotis callaar AfrIn Z xa A a tha racuarraian fufrmulaaa that Wham H U the mva1 1cianat nf J in Ta1I is 2quot 50 239 Tx has l adillg acaa acianat 1 palgmamiaalza acamplt Lat aly salve aCha b3m539hav a pmblam in the aanaaa tlhat they have the fnllawing ramaa139kabla praparaty w z Amang ah p aa 39ynamfa a aPvzI Hf dagraa n 0 with Iaadirtg cra icfa39n L 21quot Tr39 daviara 39a39aa f ffmm aam in the rarvat39 1 Q p y max P1 E39 maaa ia2iquotquotiquotJz IQJL6 v IrE 7I E lSI39Equotl Prmaf Firata the aqualmr in 16 fallaawa at aazaa from max LTgrl 5 max 105 n l 1 i n I x5I Ta 1aamplaata ma aargumaam we aasurna that aPt 5 ca palynamaial gaf tha stated type Em w hja h max EF r ii 2quot 1739 i IEquot396 POWER semras SOLUTIONS AND SPECIAL FUNCTIONS 235 and we deduce a contradiction from this hypothesis We begin by noticing that the polynomial 2quotquotTx 2 quot cos n0 has the alternately positive and negative values 2 quot 2 quot 2 quot l2 quot at the n 1 points x that correspond to 0 O Jrn 2n mtn Jr By assumption 17 Qx 2quotquot7x Px has the same sign as 2 quot7x at these points and must therefore have at least n zeros in the interval 1 lt x lt 1 But this is impossible since Qx is aquot39polynomial of degree at most n 1 which is not identically zero In this very brief treatment the minimax property unfortunately seems to appear out of nowhere with no motivation and no hint as to why the Chebyshev polynomials behave in this extraordinary way We hope the reader will accept our assurance that in the broader context of Chebyshev s original ideas this surprising property is really quite natural For those who like their mathematics to have concrete applications it should be added that the minimax property is closely related to the important place Chebyshev polynomials occupy in contem porary numerical analysis NOTE ON CHEBYSHEV Pafnuty Lvovich Chebyshev 18211894 was the most eminent Russian mathematician of the nineteenth century He was a contemporary of the famous geometer Lobachevsky 17931856 but his work had a much deeper in uence throughout Westem Europe and he is considered the founder of the great school of mathematics that has been ourishing in Russia for the past century As a boy he was fascinated by mechanical toys and apparently was rst attracted to mathematics when he saw the importance of geometry for under standing machines After his student years in Moscow he became professor of mathematics at the University of St Petersburg a position he held until his retirement His father was a member of the Russian nobility but after the famine of 1840 the family estates were so diminished that for the rest of his life Chebyshev was forced to live very frugally and he never married He spent much of his small income on mechanical models and occasional journeys to Western Europe where he particularly enjoyed seeing windmills steam engines and the like 39 Chebyshev was a remarkably versatile mathematician with a rare talent for solving di icult problems by using elementary methods Most of his effort went into pure mathematics but he also valued practical applications of his subject as the following remark suggests To isolate mathematics from the practical 3 Those readers who are blessed with indomitable skepticism and rightly refuse to accept assurances of this kind without personal investigation are invited to consult N I Achieser Theory of Approximation Ungar New York 1956 E W Cheney Introduction to Approximation Theory McGrawHill New York 1966 or G G Lorentz Approximation of Functions Holt New York 1966 0 DWFFERENTEL eetlsmeets tle mAennd s set the seiiernees te i tTquotil quottquotit the SI I iiitquot ef e sew shut ewer item the He We r39ked in imenty elds but his meets ijmgttenit eehaiiesernents were in 39prKeibe niiity the Ihreer set nuimhters end the eppmtsimetquotiten ef inrtiensi te which he was let by his interest in 1rampEIh l1i5t 5 In p 1 39iZ3 b39il i39lm he ii tf Ett fit1EE the eeneepts ef miethetne tieeI espee39tetiisn and verieneei fer sums and srititmertie means ef rendem sariisbiesi gave e beeiutifiui1ly siinpie ef the Jew out large rtunmberrs quoteased en whet nerw lenieern as Chehyshet s ineque1ity end weritedi testensiseIy39 en the eentral Iiimit t39hieeriern He is tegerLd e1 es the initeiteetuei tether est a lung series ef wetli itneiwni quotRussians seientists when eentri buted te thee rniethemietieel theeryr ef probeibility ineilming A Matthew 8 n Eiernstseirn KnIimege39re U A Kttiniczhin and e thersi In the Eaten V Chzeibtysihev 39l1elpeti te prepare en edit39ien eat serne of the werits eaf Euler It eppe t s that this tesllt Assessed to turn his ettentien te thee theerjr est ii l l1 1b6IquotE petstgiteuierhr tie the s39eIy39 di ieuiit prebiem ef the distribettieit Elf primes ts the reeder p39ebebi339 Itinmes a prime nzunvseer is an ittt ger pt 3 1 that w ne pesi39ti39ve ditritsets etteept 1 and p The i rst feew ere E 5tlI39 seen te e 2 3 5 1 13 1 19 23 29 31 53 411 It is eieer that the prirnes are distri bttted ieimeng all the pesitisret integers in e rsthaet iirregnler W jtiit fete es we metre em1t they seem te eeeer less end Itess firetquentljy end yeti there are many sdjeining peirs seipereted by e sirngie even rnimier The pretiiem ef disemreritngt the law g verning their E39I392tEl TirB tt amp Etitil of isntdierstiendinig the reeserts fer it is me that hes el39ielieingeti the euriesity ef men fer hiunttireds ef years In if 395 1 tEn1etr espressed his erwn be temei1i39t in ttquotIese ererdquots iMeti1ernetiieiens heee triied in vein te this day ten disee ser EDt39t1E erdet in the sequeinee et prinie nembzeris and we hzstrei teasein ten beifieve that it is e mf395tlEI39P39 inte winieTh the i1vLi39rnen mind wiii ineter penetrete IMenj39 tti mpt have tween made tie ndf s39intpte fiermuies iier the nth prime enti fer the ezteet zrmrnber ef mfimes emeng the first 1 pesitive integers All sueh Kefferts have feiIed and rest ptegress was i hiuE39 equotEd wniljt when metiten1stieiens ster39tted instead ts ieek for 39infermetien ebiiml the e39ut39 e t ege distri11etien ef the priirnes amen the pesiiti t39e integers It is eust ermetr ten deznete Ah rts the number ef primes less than er eque re s ptssittive inumher st Thus eI e U st2 11 It3 2 trite 2 M4 3 2 and so on In his nearly jfttth Gauss s tutdiied ttfst em pviriieellyi F the ezim ef nding e sirnpiae fLl1TIETiDi I that seems tea 39Ppt39UJtim EE it wist39h ea 5iquot a setettitre Btr f fer large I On the basis net the e eseweit39iiens he eenijee t39ureti pethaps st they age et fewteeni er ifteen that tt1f1egt is e geeczl epptrfn1ximetiittg f itInie39tittn in the sense ttihet lim W 1 C8 C8 EWI frmEt g t This steternent is then fsmees prime emeib er theerteim end es tar as enywenei knees fi ellf was never ebile te supnert his gU 5i5 with ievein ea fitegmenat eat preefl Chre byis39hes unswsre eff Cieuss s eenjettt1re was the rrst methemstieisn te ietsteI iish em l39tTl C 39FtEl siDt1iS eineut this quesitie n In 184it end 1850 he pressed 39ti39iet tis213 i e t Et r V e L ms t 19 sv39iegti POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS a for all su iciently large x and also that if the limit in 18 exists then its value must be 13 As a by product of this work he also proved Bertrand s postulate for every integer n 2 1 there is a prime p such that n lt p lt Zn Chebyshev s efforts did not bring him to a nal proof of the prime number theorem this came in 1896 but they did stimulate many other mathematicians to continue working on the problem We shall return to this subject in Appendix E in our note on Riemann APPENDIX E RIEMANN S EQUATION Our purpose in this appendix is to understand the structure of Gauss s hypergeometric equation x1 xy c a b 1xy aby O 1 In Sections 31 and 32 we saw that this equation has exactly three regular singular points x O x 1 and x 00 and also that at least one exponent has the value 0 at each of the points x 0 and x 1 We shall prove that 1 is fully determined by these properties in the sense that if we make these assumptions about the general equation y Pxy39 Qxy 0 2 then 2 necessarily has the form 1 We begin by recalling from Section 32 that if the independent variable in 2 is changed from x to t 1x then 2 becomes II I where the primes denote derivatives with respect to t It is clear from 3 that the point x 00 is a regular singular point of 2 if it is not an ordinary point and the functions 1 1 1 1 tPt and t2Qt are both analytic at t 0 We now explicitly assume that 2 has x O x 1 and x 00 as regular singular points and that all other points are ordinary It follows that xPx is analytic at x O that x 1Px is analytic at x 1 and that xx 391Px is analytic for all nite values of x xx 1Px S axquot 4 3 The number on the left side of 19 is A log 2535530quot3 0 and that on the right is A mFEnEnaT1aL39ELiM 1m45 If we sub5 tit ute 1 4 lit than 4 ecrimVes lt1gtPltgtaawltgtquot 1 Al I 1 2 39 F W 9 amp 0 y 0 r 1 r quot V4r 1 at Lq t H 3 A Since x 0 m is a reg ular singu1ar point of fumc1inn mus1 he g sin 4 3ri39ek15 analytic at I U cm 1Iclude that 1 E 213 A E ps x 1 far cesrtaain su nstan 15 A aaxnd Sim4ilarhu xzfx is anaIytic far all 50 nit amp values Of 3 sI and A 2 ha M 1 0 bnt 51 E E A5 befmre the assumprtjizm that P DD is a regular sir1gular paint Elf P implies that mum he amalrt i c at I U 90 b3 bi V 2 H and r T f V Q1 x21f 1 1 12 x Q 1 4 391 Now the fact that 6 0 ibnummd near 0 means that x3Qx bmured fm large Jr so 39 E1 5 x 1 E 1 rr 1 A ai1vsn lmund ed and C Is EL enables us tog wri te as D 0 C quot MT 393 QLI xi LI IlL3 xx 1 J and in viesw of 5 and A8 equatiDn o Atalqgs the form f rjBy P T H r x 1 J392 Jr 5 31 x 1 J1 U 9 POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS w Let the exponents belonging to the regular singular points 0 1 and 00 be denoted by CV1 and CV2 B1 and 32 y and y2 respectively These numbers are the roots of the indicial equations at these three points mm 1AmD0 mm1BmF0 mm 12 A BmDF CO The rst two of these equations can be written down directly by inspecting 9 but the third requires a little calculation based on 3 If we write these equtions as m A 1mD0 m2B1mFO m21 A BmDFCO then by the wellknown relations connecting the roots of a quadratic equation with its coe icients we obtain C1 1a 21 A a 1CY2D 1 21 B 39 31 2F 10 y1y2AB 1 yy2DF C It is clear from the rst column that 0 10 2 1 2Y1Y21 11 and by using 10 we can write 9 in the form 1quot0 1quot 0 2 1 39B1 B2 y xi x 1 y G 10 2 3132 V172 W132 5132 l 0 12 x2 x 12 xx 1 y This is called Riemann s equation and 11 is known as Riemann s identity The qualitative content of this remarkable conclusion can be expressed as follows the precise form of 2 is completely determined by requiring that it have only three regular singular points x 39 0 x 1 and x 00 and by specifying the values of its exponents at each of these points 39 Let us now impose the additional condition that at least one exponent must have the value 0 at each of the points x O and x 1 say a1 31 0 Then with a little simpli cation and the aid of 11 Riemamfs equation reduces to I WYquot 391 1 CV2 Y1 Y2 391 1xquot quot Y1Y2 0 BEFFERENTIAL eeesttews which elesrly heeemes Gteusrsts equstietn 1 we ir1treeluee the sunsh i l lfy netetieen e sq yse r 1 erg For this reesen eqnsttien 12 is sntettimtes esilleri the gerterett zed hypergeemetrie eqeettiers These results are rnerel3r the first few steps in is ferresehing theery ef dieret1tisl eqestiens initisterl hy Riemann One ef the aims ef this ethreery is to ehsrseterize in as simple s manner as pessihle slll differe ntinl equstisntns whese sehttiens are BKpIquotESS iiJiE in terms est Gsuss39s hyper geemetrie ftInetieh Anether s te sehiese s systemstire eisssifiemien ref eli di erwentiei eqns39tiens with rational eeeetiehts aeeeridittg the the number and nature ef their singuitsr peints One surprising feet that ernerges frem this elsssi eetien is that s39irttuell39 sit such equstiehs iarisiing in Instherneti eel physies eezn be generated hy E ii ll frent s single equetien with thee reiulsr singularquot peints in whieh the dierenee between the exponents at eseh point is U39t2 NOTE ON Ne greet mine ef the past has exerted s deeper iniiuenree en the n1sthemsties ef the twerltieth eentury than ernhsrri Riernenn 12e i366 the sen of a peer eeuntry minister in nerthern Germisny He stttdixierd the werhs ef Euler and Legendre while he was stili in seeenriery seheei end it seiri that he mastered Legendr e39s treetise en the therr ef rnumhers is less then in week But he was shit and medestt with Iittie swerreness his ewn evsttraeridine39ry abilities sen et the ege ef nineteen he went te the UniVETSit39 ef Gtitt39inge n with the stint ref piessicng his father has isriudying theeiegy and beeeming a minister hirnseif Fertune teijy this strewrtiw pttrpse seen stitch in his threat and with his fethefs wi1ling perrnissien he switched te rnstihernsties The pvresenee ef the iegendery Gauss sutemeti esiiy made G ttingen the center ef the ma themetiesl 39werid But Gauss was remete end urmppreeeheble v pe39rtiet1Ieriy rte heginnmg sturients end sfters eniy s year Riemann ieft this unsstisfying erwirenment ens went te the Unis39eiquotrsit3r ef Berlin There he sttrseted the fr39ienrliy interest of Dirichlet and Jeererhi and leerneti s greet desi frem heth men Ttee yveers lleter he returned t Gettingen where he ehtieined his deeter s degree in 1851 D uring the neJtt eight quotE 39113939 he endured ciehiiiitetineg peteerty and erested his greatest werirst In A1354 he was appeiinteeii Pris39stdesent unee39irri leetuirerI whieh at that time was the neeessery first step en the sesdernie iedder Gauss IiiEitii in quot155 and Diriehiet was eeiled tie G ttirtglein at his 511CCES5 39 Diriehiet helped Riemann in every way he eeuid yrst Awitih s smsii ssisry shetut enetenth net that paid re a felt prefesser and thezn with a premtien te en sssistsnt preiessership In 1359 he sise dined snri Riemann ewes eppeinted es at full ptretfesser te repisee hitre Riemenri s years of pesquoter39ty39 were esre39r Abe his hesith A full utnuzierstsndittg ef these further udesreiepntents requires s grssp ef the main ZI39Ii1Ci39piEStJ f ettmpies shsijjtsis Netrertheiess s reeder witheut this eqtIiprr1en t esn glean a few useful irttpressiens irerh E Whitttehrer and N Wetsen MeeTern Aneiysir pet 2U3 203 CeItttrid g e Uniquotrersvi39ty Press Lendeh 19353 er D Rsirwiiiieg intfermeeiEettle Dr quoterenriei Er1r39eeitiens ehep 6 Meemiiiitent New 39tquot39erk 196st rowan SERIES SOLUTIONS AND specw FUNCTIONS 241 was broken At the age of thirtynine he died of tuberculosis in Italy on the last of several trips he undertook in order to escape the cold wet climate of northern Germany Riemann had a short life and published comparatively little but his works permanently altered the course of mathematics in analysis geometry and number theory His rst published paper was his celebrated dissertation of 1851 on the general theory of functions of a complex variable Riemann s fundamental aim here was to free the concept of an analytic function from any dependence on explicit expressions such as power series and to concentrate instead on general principles and geometric ideas He founded his theory on what are now called the CauchyRiemann equations created the ingenious device of Riemann surfaces for clarifying the nature of multiplevalued functions and was led to the Riemann mapping theorem Gauss was rarely enthusiastic about the mathematical achieve ments of his contemporaries but in his official report to the faculty he warmly praised Riemann s work The dissertation submitted by Herr Riemann offers convincing evidence of the author39s thorough and penetrating investigations in those parts of the subject treated in the dissertation of a creative active truly mathematical mind and of a gloriously fertile originality Riemann later applied these ideas to the study of hypergeometric and Abelian functions In his work on Abelian functions he relied on a remarkable combination of geometric reasoning and physical insight the latter in the form of Dirichlet s principle from potential theory He used Riemann surfaces to build a bridge between analysis and geometry which made it possible to give geometric expression to the deepest analytic properties of functions His powerful intuition often enabled him to discover such properties for instance his version of the RiemannRoch theorem by simply thinking about possible con gurations of closed surfaces and performing imaginary physical experiments on these surfaces Riemann s geometric methods in complex analysis constituted the true beginning of topology a rich eld of geometry concerned with those properties of gures that are unchanged by continuous deformations In 1854 he was required to submit a probationary essay in order to be admitted to the position of Privatdozent and his response was another pregnant work whose in uence is indelibly stamped on the mathematics of our own time The problem he set himself was to analyze Dirichlet s conditions 1829 for the representability of a function by its Fourier series One of these conditions was that the function must be integrable But what does this mean Dirichlet had used Cauchy s de nition of integrability which applies only to functions that are continuous or have at most a nite number of points of discontinuity Certain 33 His Gesammelte Mathematische Werke reprinted by Dover in 1953 occupy only a single volume of which two thirds consists of posthumously published material Of the nine papers Riemann published himself only ve deal with pure mathematics 34 Grundlagen fijr eine allgemeine Theorie der Functionen einer ver inderlichen complexen Gr o sse in Werke pp 343 35 Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe in Werke pp 227264 L niFFEttEHI39tIaL EtZtUaTt0lrs hrnetineints that arise in number theerjr sugges39ted te Riemann trhat this definitien sheuld he lzvlreariened He rlerel eped the eeneept ef the Riemann irn39tegral as it new appearhs in rnest t lttti ks en ealenlns established neeessary and su ieie nt eentlitiens fer the esistenee ef sueh an inntegrel anurzlr generalized iriehfletis er iter39ia fer the 1raiirlitt ef F39eurier ettpa39r lsiens Cantefs fameus theery lei sets was direetiy inspired by a prebiem raised in this paper and these ideas led in turn 1 the eeneept ef the Iebesgne integral and even rnere general types eif integratien RieInannquots pieneering inrestigatienus were thelrefere the first steps in anether new hraneh emf tltathemarties the theetry ef T lI1C39i iCIIt5 if a real trariabie 39 The Hiernann rearrangernent theerern in the 39theery ef in nite series was an ineidental result in the paper just rleseriher He was famili ar with lJinquotehletquot39s lenameie shewitng that the sum ef a weenditienalil1F eenrergenrt series ean he ehanged hy altering the erder ef its terrns 1 1 1 s leg2 13 l 1 2 3 4 5 1 1 C 1 1 l 7 1 EI i EI39 i g2 c It is apparent that these twe series have di erent sums hut the same terms fer in i4 the grst twe eelsi ttise terms in 113 are feilewed he the first negative term then the neat twe pesiitive terms are feilewed by the seeentl neatise term and se en Riemann preveti that it is pesslihler te rearrange the terms ef Il39ljr39 E d39iti Iii39 eensergent series in such a manner that the new series will ee39nverg e lie an arbitrary quotprerigneid sum er rliW 39letrge tn W er 33 In addittien be his parehartienaryl essay Riernann was else required ate present a trial leetmer tn the fazenltgr fhefere he eenlrl he appeinterl tn his nnpairil leetnreshipr It was the enstem fer the eandiLLlate te ether three titles and the hearl ef his espatrtment usualljy aeeeprted the rst Hewevert Riemann rashly listerl as his thirrl tepie the f eundatiens ef geetmetry a preteunril subjeet en which he was unprepared hut whsieh Ciausts hadheen turning ever in his nline fer es yearst Natrnrallgy Gauss was eurieus the see hew this partieular eanrlidateis gllezrieustly fertile erigiinarlrity weul eepe with sneh at ehallenge and tn Rien1enn39s dismay he designaterl 1his as the snibijleet ef the leeture Rie39mann quieitl1r tere himself away frern his ether interests at the time quot mjr in39restigtiens ef the eenneetienr hetween eleelt39rieity magnletisrrn light and gr39atritatietn antrt wrete his lecture in the next twe nrenths The result was mine ef the great elassieal rnasterrpileeetsr ef rnathe rriatiest arnli prehahly the meet irnpertant seienti e lecture ever gi39renf quot It is reeerdeti that even Gauss was surprised and enthusitastie Riernantrs leetnre presenterl in nenteehnieai lannage a vast generalieatien ef all ltnewe geernetrries beth EZueiirllean and nenEneiidean This eid is new ea lleti Ftientarlntnian gED mBl llZ39 39 and apart frem its great irnpertanee in pure H1 Usher riie Hujrpethesen Weliehequot tier Geernetrie an i39iru1tltieliegen in lviquoterulre Apps 239i 2v 239E r There is a trnnslatien in U 2 Smith SC39l fEE see in lili if 3fhEI ri3tilC3 Me rawHii1l hlew tquotquotet tt 1929 rowan SERIES soumons AND SPECIAL FUNCTIONS 243 mathematics it turned out 60 years later to be exactly the right framework for Einstein s general theory of relativity Like most of the great ideas of science Riemannian geometry is quite easy to understand if we set aside the technical details and concentrate on its essential features Let us recall the intrinsic differential geometry of curved surfaces which Gauss had discovered 25 years earlier If a surface imbedded in three dimensional space is de ned parametrically by three functions x xuv y yuv and z zuv then u and u can be interpreted as the coordinates of points on the surface The distance ds along the surface between two nearby points uv and u duv dv is given by Gauss s quadratic differential form dsz Edu2 2Fdudv Gdvz where E F and G are certain functions of u and v This differential form makes it possible to calculate the lengths of curves on the surface to nd the geodesic or shortest curves and to compute the Gaussian curvature of the surface at any point all in total disregard of the surrounding space Riemann generalized this by discarding the idea of a surrounding Euclidean space and introducing the concept of a continuous ndimensional manifold of points x1 x2 x He then imposed an arbitrarily given distance or metric ds between nearby points xxZx and x dxx2 dx2x dx by means of a quadratic differential form 132 2 81 dxi dxir 15 ijl where the g are suitable functions of x x2 x and different systems of g de ne different Riemannian geometries on the manifold under discussion His next steps were to examine the idea of curvature for these Riemannian manifolds and to investigate the special case of constant curvature All of this depends on massive computational machinery which Riemann mercifully omitted from his lecture but included in a posthumous paper on heat conduction In that paper he explicitly introduced the Riemann curvature tensor which reduces to the Gaussian curvature when n 2 and whose vanishing he showed to be necessary and sufficient for the given quadratic metric to be equivalent to a Euclidean metric From this point of view the curvature tensor measures the deviation of the Riemannian geometry de ned by formula 15 from Euclidean geometry Einstein has summarized these ideas in a single statement Riemann s geometry of an ndimensional space bears the same relation to Euclidean geometry of an ndimensional space as the general geometry of curved surfaces bears to the geometry of the plane The physical signi cance of geodesics appears in its simplest form as the following consequence of Hamilton s principle in the calculus of variations if a particle is constrained to move on a curved surface and if no force acts on it then it glides along a geodesic A direct extension of this idea is the heart of the 3quot This is proved in Appendix B of Chapter 12 S DFFE EI iT ML IEEiU ATl H5 gennessi tiheerry ef reietivitn3r whiehn is ierssentigsiiy is tihenry nf gsrsvits Isinein Einsteiin enneeis39ed itiie ge e n1re tr3r ef spasee as a Rie39n1isnn isn geometry in whieth the euirvstnure and geiecieisizes are detern1inenl by the distiriutien etf 39 39l3lH ET in this cursed yspecie planets zmeee in their nzrbirts around the sun Abs sin1pIi3 eesstiing einzmg gendesies inisitesd sf being peiieid ninm einrved paths by s m 1stesiE u5 fezrce ef gravity whesse nature ne me hens ever really inndersL nL E In 1359 Riemann pubIished his only weri39 en the tiheery ef niumbers a brief but erzeeieidinghr prefeund pepegr ef less than 110 pages devetsed lie the prime number theer39eimn339 This 39mighity effefrt started tie waves in ismrersi branches of fFl E7lquotB methemntimi and its ini nenes will prebsflzaslyr stii be Visit a thnnisasnd years from rmw His starting paint was at remns rksbiIn e findeintnity diiseosrered by Euler Inner ianC 11EiLIquot earlier if squot is as real nnurnber greater than 1 39Il1en 1 in 1 gt 39 ills where the esipressien en the right denntes the preduet ef the numbers 1 n p quot 39 fer all prinnes p Te understcand hew iihi idenIity arises we nets that 1r ii is E 11 i 1 N fer Lry cs 1 sn fer sash p we shave rn n1nuilrEipiying tiiese seiries fer sill primiesi p and retailing sihst eselhn iyntveger H Ti is 39uniq1uIeiy39 espressible as an rediinet of powers nf di erem primes we see that W 0 X X p1 mp P p p Ii Ii Illlll I I is 3 r l whieh is the iderntiity 16 The sum ef the seiies en the iefi mi 116 is veviden39tii3r is funeti en of then real squot i iaiiE s i39 1 and the 39id39ELquotIitiEquot estsniblisi1esi e eenneietie n between the beimsrier ef this function and prepertiees of the pi imes Euler himself espI eiIed this eenneetien in several ways but Riernsnn peree39ived that seeess tn the deeper fEVa IEU r39EIE ef the diistributien sf pri39rnes can enmity he gained pN aiiewing 5 ts be H ieernplei2 39sssieb Ie He deneied the res39nltinsg funetien by 39If sl send it hes sinsei been knnwin as the REemenn sens furfmrfen isjI 1 s39anin39 1i TI 1 iL e39ner dine Aneshli tier PrinislsMen umeur signer gegebsenen Curiiss e in Wsrskei pp iidfl s15i3 i See izhze 5iJ 391TtiETE1lEn39l nf Iifhe prinie LI1 iJ39I3939Ib391 i 39Ehem em in nus neie inn C39hehrshev in ipp envd39ins D power SERIES soumons AND SPECIAL FUNCTIONS 245 In his paper he proved several important properties of this function and in a sovereign way simply stated a number of others without proof During the century since his death many of the nest mathematicians in the world have exerted their strongest efforts and created rich new branches of analysis in attempts to prove these statements The first success was achieved in 1893 by J Hadamard and with one exception every statement has since been settled in the sense Riemann expected This exception is the famous Riemann hypothesis that all the zeros of Cs in the strip 0 S 0 S 1 lie on the central line or It stands today as the most important unsolved problem of mathematics and is probably the most difficult problem that the mind of man has ever conceived In a fragmentary note found among his posthumous papers Riemann wrote that these theorems follow from an expression for the function Z39s which I have not yet simpli ed enough to publish Writing about this fragment in 1944 Hadamard remarked with justified exasperation We still have not the slightest idea of what the expression could be quot He adds the further comment In general Riemann s intuition is highly geometrical but this is not the case for his memoir on prime numbers the one in which that intuition is the most powerful and mysterious 39 Hadamard s work led him to his 1896 proof of the prime number theorem See E C Titchmarsh The Theory of the Riemann Zeta Function chap 3 Oxford University Press London 1951 This treatise has a bibliography of 326 items 4 Werke p 154 4 The Psychology of Invention in the Mathematical Field p 118 Dover New York 1954 E5 D NS pT A E FTLIRIER T139igD11 mEI39iC series of tl e farm f x gag d Ann cogs b sin G 1 are EEdEd in 1116 treat mem f manyr phgytsicall iplf bIEME that lead rm par4t ia di FerampI1 tial iqua tim1s for izmtance in the thenry of snund heat mnductinn elemmmAagnetic waves and mzechanmal vibratimml We shall Examine same of these ajpplicat iDns in the next Eh ptIB7E The Impre 5eVntati 3nV vauf functirJn4s by Apmwer seriEs is familiar ta us fmim calculus and aim f1gm our wile in the pEECEdi g vl1ap ter An impm tant advant1ageJ of the serie5 1 is that P T nan I39E pI E5Ei l very gampneral Vfunctiin5 with many isumnt4inuitVia51iIee time di5mntinu aus A impulse fujmtions f electriral eninVeering whereas pnwer 5erie5 can represent onl r c m1tiVnutm5 func ti ns that have derivastives Anf all m39ders It is nnl f1 reasmzs of Ii39ifZquotI39I lquot rquotIE39E II7 lZr 3 Ihat the mnstart I rm in H is wrimn at in539tead at am This will bEc lll u IiaaLar bslnw p G FOURIER SERIES AND ORTHOGONAL FUNCTIONS 0 Aside from the great practical value of trigonometric series for solving problems in physics and engineering the purely theoretical part of this subject has had a profound in uence onthe general development of mathematical analysis over the past 250 years Speci cally it provided the main driving force behind the evolution of the modern notion of function which in all its rami cations is certainly the central concept of mathematics it led Riemann and Lebesgue to create their successively more powerful theories of integration and Cantor his theory of sets it led Weierstrass to his critical study of the real number system and the properties of continuity and differentiability for functions and it provided the contextquot within which the geometric idea of orthogonality perpen dicularity was able to develop into one of the major unifying concepts of modern analysis We shall comment further on all of these matters throughout this chapter We begin our treatment with some classical calculations that were rst performed by Euler Our point of view is that the function f x in 1 is de ned on the closed interval r lt x s yr and we must nd the coefficients a and b in the series expansion It is convenient to assume temporarily that the series is uniformly convergent because this implies that the series can be integrated term by term from 1 to 1132 Since 1 I cosnxdx0 and J sinnxdxO 2 1 for n 1 2 the termbyterm integration yields I fxdx non 30 no 31 fxgt dx 3 It is worth noticing here that formula 3 shows that the constant term ao in 1 is simply the average value of f x over the interval The coefficient a is found in a similar way Thus if we multiply 1 by cos nx the result is S fx cosnx cz 0cos nx l a cosznx 4 2Readers who are not acquainted with the concept of uniform convergence can freely integrate the series term by term anyway as Euler and his contemporaries did without a qualm as long as they realize that this operation is not always legitimate and ultimately needs theoretical justi cation DIFFERENTlsL EG39UATmN5 where thze terms nat Vwritxten vnmtain prmjiuctJ5 of the farm sin c n1swrL Ell of the farm ms H ms H1 P3 1 At thxis paint it is nEcequotss ampry In recal l the l139ig11 metric identitie5 sin m sirm H nx L105 H ms cns m z nVx chews E nvr sin F sin sir1 mix Hans m rIx ms w r1x whim f w 39 2iIB5Cty fmm the additinn and S11lJlI39f3EIiEiII f mrmu a5 fmf the sine and msine 11 is now xeasfy to verify39 than fur integral valrules of m and H 3 1 we have I sin E casts U 5 and J t1I39EC S Z it DJ m 5 Hi 6 These flacts Enabblie was In integrate 14 tErm by term and btain pb 1 J Jr crasynx an I cnslnxdx P nit a pg SE a i E f 15 C05 T J By 3 formula 0 is alsn ra iVd1 far 1 2 0 this is the reason fem writilng the t0n5taI1t term in 1 as an rasthcr than am We get the corrV esp0ndVing fmmula for b by VerssentialJly 39IhE same p1r r4 dureEwc multiply 1 thrnugh sin wax i EgIquot l term by taerm and use the add itiunaIl fact that I sin sin p in PU i we 8 K51 yiclds J fgx sin P dx b J T mg mg 2 2 i rm sin rm 08 9 Th sgte ca culaft iUn5 shnw lat if the series is unif7m39VmI39 cs nvarm ernt than the cuef cient5 an and hm EE1iI39N he nbVtaignd frnm the mm f3939 by means of the abme fm mulas HnwVcver this SitLlEti I 1 is man IfEStf39iiE t d ta he Uquotf much prac1ticVal valtnia becausE hw Jim we kn w Wh lih f 3 given f admi15 an expan3i n as 3 unimrmly c0nverent Il1figDnmmamptVric serAie5 FOURIER SERIES AND ORTHOGONAL FUNCHONS 249 We don t and for this reason it is better to set aside the idea of nding the coefficients a and b in an expansion 1 that may or may not exist and instead use formulas 7 and 9 to de ne certain numbers a and b that are then used to construct the trigonometric series 1 When this is done these a and b are called the Fourier coe icients of the function f x and the series 1 is called the Fourier series of f x A Fourier series is thus a special kind of trigonometric series one whose coef cients are obtained by applying formulas 7 and 9 to some given function f x In order to form this series it is not necessary to assume that f x is continuous but only that the integrals 7 and 9 exist and for this it su ices to assume that f x is integrable on the interval 71 S x S n3 Of course we hope that the Fourier series of f x will converge and have fx for its sum and that therefore 1 will constitute a valid representation or expansion of this function Unfortunately however this is not always true for there exist many integrable even continuous functions whose Fourier series diverge at one or more points Advanced treatises on Fourier series usually replace the equals sign in 1 by the symbol in order to emphasize that the series on the right is the Fourier series of the function on the left but that the series is not necessarily convergent We shall continue to use the equals sign because the series obtained in this book actually do converge for every value of x Just as being a Fourier series does not imply convergence conver gence for a trigonometric series does not imply that it is a Fourier series For example it is known that sin nx 2 5 10 g 1 n converges for every value of x and yet this series is known not to be a Fourier series This means that the coefficients in 10 cannot be obtained by applying formulas 7 and 9 to any integrable function f x not even if we make the obvious choice and take f x to be the function that is the sum of the series These surprising phenomena prevent the theory of Fourier series 3In this context integrable means Riemann integrablequot which is de ned in terms of upper sums and lower sums and is the standard concept used in most calculus courses 4For convergence see Problem 2a in Appendix Cl2 of George F Simmons Calculus With Analytic Geometry McGrawHill New York 1985 The fact that 10 is not a Fourier series is a consequence of the remarkable theorem that the termbyterm integral of any Fourier series whether convergent or not must converge for all x and this is not true for 10 251 FOURIER SERIES AND ORTHOGONAL FUNCF IONS v Al I bl 1 p I I I Zn 0 4 p FIGURE s3 It is easy to see that each term of the series 11 has period 2r in fact 27 is the smallest period common to all the terms so the sum also has period 21 This means that the known graph of the sum between 75 and Jr is simply repeated on each successive interval of length 2 to the right and left The graph of the sum therefore has the sawtooth appearance shown in Fig 33 It is clear from this that the sum of the series is equal to x only on the interval 17 lt x lt 15 and not on the entire real line oo lt x lt 00 It remains to describe what happens at the points x in l3r where the sum of the series as shown in the gure has a sudden jump from 1 to Jt By putting x lr l3Jr in 11 we see that every term of the series is zero Therefore the sum is also zero and we show this fact in the gure by putting a dot at these points The rst four terms of the series 11 are 2 sin x sin 22 sin 3x sin 4x These and the next two terms are sketched as the numbered curves in Fig 34 The sum of the four terms listed above is y 2sinx sin2x 23sin3 sin4x 12 Since this is a partial sum of the Fourier series and the series converges to x for r lt x lt 71 we expect the partial sum 12 to approximate the function y x on this interval The accuracy of the approximation is indicated by the upper curves in Fig 34 which show this partial sum of four terms and also the sums of six and ten terms As the number of terms increases the approximating curves approach y x for each xed x on the interval 1 lt x lt 717 but not for x in Example 2 Find the Fourier series of the function de ned by fx 0 fx In rSxltO Osxsyr n1FFE39HEMT1AL EQ UnT1DHT5 p I Pv Ten IvEE SM lien11 Fuur HIquotm2 V P By 3L fr and quot9 we ehave an I FEdl Ii 71 V i an J 1 c t35nx D H E 1 Hquot In 31 U b IE sin E p u H ill 7 I H f I I 3 ll 31 n 1 Since the nth ewsn 39iLi mhE139 is 2n and th nth add nuxmjber is Zn 4 the East HIT these fm rquotmu Iias taa s us that b39m V U ban1 Hy sub3tituti4ng tin 11 we 39b39EIin the reiquirEd Fi11riaEr series f L1 25inx p Sigir P x W3 FOURIER saunas AND ORTHOGONAL FUNCTIONS 253 FIGURE 35 lr39 hIjlt jg c4 The successive partial sums are Zsinx y 33 Zsinx sin3x NI y 27 y The rst four of these are sketched in Fig 35 together with the graph of y f x We will see in the next section that the series 13 converges to the function f x on the subintervals 1 lt x lt O and O lt x lt Jr but not at the points 0 Jr Jr The sum of the series 13 is clearly periodic with period 21 and therefore the graph of this sum has the square wave appearance shown in Fig 36 with a jump from 0 to It at each point x O in l2r Further this sum evidently has the value 312 at each of these points of discontinuity and we indicate this fact in the gure as we did before by placing a dot at each of the points in question And just as before each dot is halfway between the limit of the function as we approach the point of discontinuity from the left and the limit from the right quotll I I i i i 3 H I I 1 F l E n i i 2 quot i i i i J L 39 L 39 E 4n 311 211 n 11 Zn 311 tin FIGURE 36 115 DIFF ERiE HTiAL iEEQ JiliTTiU39i 5 Example iFiindi the Fourier l5 EvTirE39 nf the fu nE IliD ClE I39TIEdi by 175 im5J1 lJ mm I J I I4 m gxim This is the ifuinmioin in fExamjpie pj m irms the E nstant iaquot2i Its Fourier 5arias can therefnrie be btia39ined by suibitractiing nu 39frUm the SE1 iE5 13 w hiiri39r1 giiquotiquotE sin 3 sin 5 T U 5 14 x graph of the sum Df this L ts is simply the square Aways in 36 iifznwered in quotbe sirmm I ci b ul ti1aeua aJtiSv gsimwn in IE ig Eii mple M Fi id the iFuurier EEIquoti S of the funcitiiun de inad by Ex am 5 1 vi ill I 31 fix ha fix p Ex 0 5 x l This is iii func nin wie ine di in E3iample 3 minus ninehiaif the ifiunimiun in Eiamplei 1 F39miiquotier series can quottherefore be attained by 5ubtracting I1E hEifV irfhe seri e5 114 term bf mrm Emm the isEris 14 sin 3 sin 3 H 5 sinimx E The gmgpih of sum anf this series is the S W39U Ih wave shown in y sin I 5 III sin J Halal H J J ifji 1 3 FOURIER snnuas AND ORTHOGONAL FUNCTIONS 255 4ni 2n1L quot39 K MI I 1 F 4 1 ltgt FIGURE 38 The validity of the procedures used in Examples 3 and 4 depends on the easily veri ed fact that the operation of forming the Fourier coe icients is linear that is the coe icients for the sum f x gx are the sums of the respective coef cients for f x and for gx and if c is any constant then the coe icients for of x are c times the coe icients for f x Also the Fourier series of a constant function is simply the constant itself Remark 1 In Section 36 we show how the interval 2 lt x 5 Jr of length 2Jr can be replaced by an interval of arbitrary length with no difficulty except for aslight loss of simplicity in the formulas This extension of the ideas is necessary for many of the applications to science Remark 2 Our work in this section and throughout this chapter rests on the property of orthogonality for the system of functions 1 cos nx sin nx n 1 2 over the interval 1 S x lt Jr This means that the integral of the product of any two of these functions over the interval is zero which is precisely the substance of equations 2 5 6 and 8 We shall return to this concept in Sections 37 and 38 and use it to give a simple and satisfying geometric structure to the theory of Fourier series NOTE ON FOURIER Jean Baptiste Joseph Fourier 17681830 an excellent mathematical physicist was a friend of Napoleon so far as such people have friends and accompanied his master to Egypt in 1798 On his return he became prefect of the district of Is re in southeastern France and in this capacity built the rst real road from Grenoble to Turin He also befriended the boy Champollion who later deciphered the Rosetta Stone as the rst long step toward understanding the hieroglyphic writing of the ancient Egyptians During these years he worked on the theory of the conduction of heat and in 1822 published his famous Th orie Analytique de la Chaleur in which he made extensive use of the series that now bear his name These series were of profound signi cance in connection with the evolution of the concept of a function The 12516 nIFFEn39EnTinL E UA11 H 5 general sttitnde an that time was en eslii fs is fnnetien y it ennid he represenlten by s s ingiei enpressinn likes a peiynelmisL e nite een1fhinne tinn sf esiInen1nennetnsy fnnetinnsn a newer series En esx j er s n39igenen1estsrie series ef the term 0c need ens sin an if the girsph ef fx were quot erhinsry fer ienarnpie s pailyegensi line wish a number of ennners and even a few gsspzsathnn x wnulld net have been seeepieni as a genuine fnnetienz Feurier eilshned that i es bitrsjy graphs een he represented hy tnreigene meit e series end shnnid thseretinse he trfes1ecl es iegitinn1te efnnetinns and it sense as a shank In men that he Inrned nut me be righ tn It was a ieng time befnrne these issues were eemnpaleteiy elsri ied7 and it was nee eecident that the definitions ecf he funetien that is new slmes t unis e Ely was lrst fermulsted by Dinehiieet 0 V133 in n resentreh paper en the theory ef Fetnsier S i i Aisn the classical aie nnitirnn sf the ele nifIe insteygrsi dine in Riernsnn was rst given in his fnndenienntsi paper nf 1354 en the snhjVee te etf Fnanries series Indeed men ef the Inesquot irmner39tssrn mathematics rziisensreries ef the nineteenth eenntnry ere deireieti 39 linked tn the theer39y if Fnnrier series and the sppiiiimiiinnnts elf this snhj eet tn Instnhenn1stiesLl physics hssre been seereeiy less preifeundn Fn nrier nhin1seif is nne in We fengnnnetie fewz nsimie fbeenme rented in eii eiise lis eti inngneges as an sdjesetine that is weli knnewn tn nhynsiessi seiennnsts and n1nthen1stireiens e vesy en nf the World QROTBLIEMIS Find the Fnurier series fer the fnnettinn de ned by In if j l j fi l Find the Fnrier series fer the fn nnetien de neci by x ii 0 47 E s 1 D fslt 1 E l s j 8j p g J 155 J1 r 4 Eipivs Prnizriem 3 with sin is repieeen by C051 quote quot FOURIER SERIES AND ORTHOGONAL FUNCTIONS 257 5 Find the Fourier series for the function de ned by a fx 71 7 S x 5 Jr b fx sinx n39 S x S 71 c fx cosx 1 S x S 71 d fx n sinx cosx 1 5 x 5 Jr Pay special attention to the reasoning used to establish your conclusions including the possibility of alternate lines of thought Solve Problems 6 and 7 bygusing the methods of Examples 3 and 4 without actually calculating the Fourier coefficients 6 Find the Fourier series forthe function de ned by E a 39x a 1 S x lt 0 and fx a 0 s x 5 Jr a is a positive number s l 1 n39 lt x lt O andfx 1 0 s x 5 11 cf fx 1 52 lt 0 andfx 3 0 lt x lt 71 d fx 1 1 5x lt O andfx 2 0395 x lt Jr 6 1 Jrsx lt0andfx 2 Osx sin 7 Obtain the Fourier series for the function in Problem 2 from the result of Problem 1 Hint Begin by forming Jr the function in Example 2 8 Without using Fourier series at all show graphically that the sawtooth wave of Fig 33 can be represented as the sum of a sawtooth wave of period at and a square wave of period 21 34 THE PROBLEM or CONVERGENCE The examples and problems in Section 33 illustrate several features that are characteristic of Fourier series in general and which we now discuss from a general point of view Our purpose is to attain a good understanding of a useful set of conditions that will guarantee that the Fourier series of a function not only converges but also converges to the function We begin by pointing out that each term of the series 1 x fx Eao Z a cos nx b sin nx 1 1 has period 211 and therefore if the function f x is to be represented by the sum f x must also have period 2tr Whenever we consider a series like 1 we shall assume that f x is initially given on the basic interval Jr 3 x lt 71 or 1 lt x spar and that for other values of x fx is de ned by the periodicity condition fxi 2 fx 2 i In particular 2 requires that we must always have fx f Jr ntjaaanawrtat EG39U T t39U HS Aeeetrdiintgiy the eemtpleite fuinetintn we eensider is the seecalled quot periedie extensieni ef the etilgitnally given part ate the suteessive initervalis ef ietngth 2 that lie to the tight and left at time basic intewaii The phrase simpie t SC nIftMfi39 er eften jump d7isteantiniutiEi is need ten deseribet the situatiinn wl1ere a tanetient has a nite jnmp at a paint 1 x This means titat if appmaetlesz nite nit di erent limits tram the left side at x and item the trightt side as slmwni in Fig 39 We eat eexptess this Tbethavinr by wtiiting 1 ilijm fx E e um fst M39 E E 2 0 EI3 5tquott o I where it is understand that hath limits exist and are nite It will be eenmenieent ate tlenete these limits by the simpler wmbeiis ftg and f 9 set that the above inequaiity ean be written as f393539n 3 5 fists A fnnettinn f is said tn be bene ted if an inequality at the 0 if 139i 5 M helds fer some ennstan39t 7 and all 7 t1iI39IJdEIquot eenisidetateien F0139 example the tunetiants 1 e39 and sin 1 are bennded en 1 5 1 as but f 1f139t x is net It ean be pteved see Prebietm latelew that if at limanded fnnetien ffx has nanly a nite number nf diseentinnities and eniy a nite number at rnaxi ma and ntiinienia then at its diiseentinnities ej A i Ttt i fixquotFit 3 H1 FIGURE 3399 FOURIER SERIES AND ORTHOGONAL FUNCTIONS 259 are simple This means that f x and f x exist at every point x and points of continuity are those for which f x f x Each of the functionsshown in Figs 33 36 37 and 38 satis es these conditions on every nite interval However the function de ned by fx sin x 39 O fO 0 has in nitely many maxima near 1 O and the discontinuity at x 0 is not simple Fig 40a The functions de ned by 1 gx x sin x 5 0 g0 0 and 1 hx x2 sin x ah O h0 0 l39A I 3quot WW e j gt1 FIGURE 40 lIlFEEi7HEsHT s39L E UATIOMS bath are canstimsunus at 1 c w39h sreass manly h gs q di ssntis blss at this paint We are new is s p 395ifti 39 to state the fmlswismg thssrsms which sstsblsishss EIE dsssirsd ssnssrgssnsss bBh 39H i f fa Vs very Ilsrge classs sf functimiwzns j slssv ilmswe in ni tely39 mam msssimss near 1 2 0 Figs 4i b and 4 s but Ilirislztiefsfs Ths rssm Assume Hm f x is de ned and bsrunzdsa far 1 5 1 r s msd r f shut it p arm a nzirss msmrEJsrsfa5I1sss39wsIEnuir ss and sniy nffgs nu mbsr of and smEnims an shals inrsrvsi LEI f x be iE HEd for amsr trsfuss sf 1quot by me psrfisdfsit sasngd iians fs 25m x Then ms FaurEErs series sf fix ssn1ssrgsss IE rm E ft 131 Hf vs39u sry39 pvsulrsr gs and I39hsrsfsr s39 if srsrsvErgss I0 fLs am sussry pain if ssn nuEry sf rm fssrrs ss 0 if at wsry Apsirir sf d scs s ns39 nu39Lry ms Us us of ms umcrfun rsa39s nsi as shes mirsimggss srf sits fuss LHE5EdE 39 Hmii I hs39rs fix sfs I T fI then 1913 Fisuirisr ssriss rs prsssnrs t ss funmi n svsryHhsrs 1quot39 Ths strmditisnnssnss simpnsssd an f x in this sthsmsrm srs csfl sd Dsirichfez sssLndEtisns sftsr thus Germans mathsmssticians P DirichIst Wh dissssmrssrsd the theorem in 12 In Appiendix A we zestabllish the same ssnclusisnn usndessr siligshstly di srssmt h j 39p fthESES ipiEE WiSE smmcntsh n essAma quotwhich are stiIIl s u cisn y weak 110 savers sslm st all spplicsstiJnnssT Ths general ssitusastisns is as fnllwswsz Thvs mntsisnuistsy of s f39uncti0n is met su ssissnr far the scsenvsrgassnss of i1s Furusrisr series In the functims and neithsrs is it nescsssssrsr That is it is quits possssiJl1le fur a discsuntismuanuss fusncvti vn tn hs rsmrssssntsd sssrywshers by its sFnuri ss ssrisss pJrsvidsd its iSEEI IIiii Ui l39iES are rssl1ativsI3r mild and psrsidssds it is anslstiss lys well 395 We remind thss rssdetr that the value If sn imstsgrsbls IE39u n ticI n can ht redss nssd at sn39 nite zrmumjler sf paims withnut shsngisng thus ss us of its imegursh snud thvsssE39srs wquotilh u t chssnging thus Faturisnr ssrriszs f the f UHCITiJ39l39L Prnfs of irishst39s ihssUrsm in s s ghmyr rrmrI gsnszrsi quotFsrrn ssn bus IFsund in 0 Q Ti7tchmssrshs The Thenry sf EM E 3939 H5 Ed sd OEfCiurli Usn is39ssrssiEjs39s Pssss WSEI pp sllil 44U73 ixn P Rtsg 5i skampi Fimsrsisr Ssrir3 s Cih m l la New Y L W5H pp 2 T43 srnI1 in H ls 393zN E339I xrrIfF 7 f d u THIJ 1i rquot a quotErsJ39 sF i L i H39T a39rn1quot39 339rEhEJg n 39 EIjquotJs PL539 PE5 C139ssf rd Uniwsr s it Press I pp 3 W12 It is p t msjm uznssnllved pr 1lI exm M ma39ihsma39Ltic39s IE nd ssnditsisns ms srs hissh nssssssry and su iICiant FOURIER SERIES AND ORTHOGONAL FUNCTIONS 0 behaved between the points of discontinuity In Dirichlet s theorem above the discontinuities are simple and the graph consists of a nite number of increasing or decreasing continuous pieces and in the theorem we prove in Appendix A the discontinuities are again simple and the graph consists of a nite number of continuous pieces with continuously turning tangents Example Find the Fourier series of the periodic function de ned by fx0 Jrsxlt0 fxx 0SxltJr First we have 1 quot lxzl Jr anJtfo xdx2 io539 For n 2 1 we integrate by parts to obtain 1 quot 1 xsinnx cosnx a xcosnxdx 2 Jr 0 Jr n 21 1 0 1cows 1 n1quot 1 Jrnz so dzquot 0 and a2 quot quot quotquot39gquotquotquotquotquot Jr2n 1 Similarly 1 1 If 7 F xsinnxdx xCOSnx Smfx Jr 0 Jr n n 0 i Jrcosmr 1 quot39 E Jr n n The Fourier series is therefore Jr 2 cos Zn 1x sin nx 1 quot1 3 re 4 W2 2n12 n 0 By Dirichlet s theorem this equation is valid at all points of continuity since f x is understood to be the periodic extension of the initially given part see Fig 41 At the point of discontinuity x Jr the series converges to fr frr 5 When x Jr is substituted in 3 this yields the following interesting sum of the reciprocals of the squares of the odd numbers 1 1 1 1 H2 2 1 33quot39 8 4 l P JIss ssssH39 rsmssL E 39U 39Tl Is15 w A A 41111Vquot J 39hJ uiI 3 H T H 311 pT 41 The same sum is sbtsinsad subsssitu1ing the paint sf smfitirnsuisty I D intn 3 FurtLhsr we can uses l IIas nd the sums of Elms r ssipross5ls sf the squares sf sh ths pssiitive intsgsr s i1 1 I e 39 5 All sths t nasdad sstsbilirssh Ems is is writs 4 1 12 d s E Pquot Z 3 311 s j 5 quot5 39 HEquot E 2 6 T1113 sum 5 was fsund by sEsEsr in s1T3E1 sn is ns sf the must msmsssbls xiiiscmrsriss in the sarzly histi ry sf in snits sesrisss Ester Gustav Lejsuns Dssirishlsts s13I3I5E1E5 94 was as Gsrmsrm mssthsmsstisisn swim made 39msnfy sm39tributi ans sf Iss ng value ts snalyssis sins r1I1mAIsf 1f hsrfr5n Ass as young man he was drsssr1 ts Paris by iths rspvutstim1s of Cauchy Fsurissr sud quotLsgstadrs but has was mass desphr in usncsd T quothis ssnssumss ssnsd Iifwsl ng snntsst w39ifth Gsussfs pK Asr hmss Ecs 13l39Jl This prsdigimus but cryptic war smfulzsinsd many of the great smsst sr s far rssching disscnrsriss in nuimssr thsszry but it was u ndsrsstDIsds by vsry few msthsmstsisiasss as fhsfc times lstisr said DirirhIst wsss mt satias s ts s1tusdys Gsuss s DsEsquEsi snes oncs 39I39I quot39Tlquot ssssrsl times but snntinsussds throughssut his life to kssp in class Izuzmczh wim the wealth sf deep mathematics thsugms which it cuntsinss PV psmsfinjg it sgsi11 and sgsin Fm this rsssrm the husk was nsvstr pm an ms shelf but hsd sn ssbi ding piss sun the tsbis st whisshs he wsrksd Dir ishsLst was the rsst suns when nst 0 nly 1l13r ssndsrstmsd tlhsiss wmquotk slsus made it ssssssi39bls is ssthsrs In later life Dirichlet tbsscams s frissnd sud disciple sf Gssuss sn d slss s frissnd and adsissr sf Riems nn sflmm he helped in s sssms sway 9 l39iSsi11irUEtv I 3ampl dissssrtstim1 In 1355 after lssturmg at Berlin for mssn5 ysass he ssssssssdsd Gauss in the p f quotfE5E EEh 1 P st Gs ttisnsgsns quotFur Esulgs s swns wanssrfuJ l3r ingen39iDus was of disisssssssring 5 sues Appssndis 0 in the Eimmsns bKs IL sitsd in Esst ns tlts 4 FOURIER srannas AND ORTHOGONAL FUNCTIONS 263 One of the Dirichlet s earliest achievements was a milestone in analysis In 1829 he gave the rst satisfactory proof that certain speci c types of functions are actually the sums of their Fourier series Previous work in this eld had consisted wholly of the uncritical manipulation of formulas Dirichlet transformed the subject into genuine mathematics in the modern sense As a byproduct of this research he also contributed greatly tothe correct understanding of the nature of a function and gave the de nition which is now most often used namely that y is a function of x when to each value of x in a given interval there corresponds a unique value of y He added that it does not matter whether y depends on x according to some formula or law or mathematical operation and he emphasized this by giving the example of the function of x which has the value 1 for all rational x s and the value 0 for all irrational x s Perhaps his greatest works were two long memoirs of 1837 and 1839 in which he made very remarkable applications of analysis to the theory of numbers It was in the rst of these that he proved his wonderful theorem that there are an in nite number of primes in any arithmetic progression of the form a nb where a and b are positive integers with no common factor His discoveries about absolutely convergent series also appeared in 1837 His convergence test referred to in footnote 4 in Section 33 was published posthumously in his Vorlesungen fiber Zahlentheorie 1863 These lectures went through many editions and had a very wide in uence He was also interested in mathematical physics and formulated the socalled Dirichlet principle of potential theory which asserts the existence of harmonic functions functions that satisfy Laplace s equation with prescribed boundary values Riemann who gave the principle its name used it with great effect in some of his profoundest researches Hilbert gave a rigorous proof of Dirichlet s principle in the early twentieth century PROBLEMS 1 In Problems 1 2 3 4 6 of Section 33 sketch the graph of the sum of each Fourier series on the interval 5Jr S x 5 Sn Use the example in the text to write down without calculation the Fourier series for the function de ned by fx x Jrltx lt0 fx0 OltxsJr Sketch the graph of the sum of this series on the interval 57 5 x 5 Sn 3 Find the Fourier series for the periodic function de ned by fx 117 11 sx lt 0 fxx 0ltxlt7t Sketch the graph of the sum of this series on the interval 5ar S x S 511 and nd what numerical sums are implied by the convergence behavior at the points of discontinuity x 0 and x Jr x FOURIER SERIES AND ORTHOGONAL FUNCTIONS 265 FIGURE 42 35 EVEN AND ODD FUNCTIONS COSINE AND SINE SERIES In principle our work in the preceding sections could have been based on any interval of length 271 for instance on the interval 0 S x 5 21 However the symmetrically placed interval 1 lt x lt J has substantial advantages for the exploitation of symmetry properties of functions as we now show A function f x de ned on this interval or on any symmetrically placed interval is said to be even if fx fx 1 and f x is said to be add if fx fx 2 For example x2 and cosx are even and x3 and sinx are odd The graph of an even function is symmetric about the yaxis as shown in Fig 42 and the graph of an odd function is skewsymmetric Fig 43 By putting x O in 2 we seethat an odd function always has the property that A FIGURE 43 iatsiFEnEr1aL E39EI1MTIv 39HSa fm D It is clear fmm the gures that I fx 2 fxdx fx Even 3 V a IIEI A amrl I my u 0 if fx 4 b cause the iintegrals r epre5er1A7t the algebraic rsignEc1 areas under tllie cfunreis These Afacts can 3150 be e5wtaIbIished by anal39yJtic reasm1in g based an the dE itinvns 1 and 2 see l Prmblem 3 bel w Pmduct5 Elf Aeven and oddi func nns have thc simplt pi CIpElquot E5 evenmren eiven EVE dd39 E add 1dd dd even which ucIrres pnn d 10 the rfamiliar FMES 1L 17 11 S1 a1 1 1 Fm instance tn prnrve the secmnd pmpertr we c nsiVder the functiml Ac fg wlwre fx is Men and J is U Then 0 w w which Sh WS that th pmduut fjnix is B Cu flazr tww pmpe rtEEs can be prveed 5imilarljr As an Example we know that 393Ga5nx is add becaust x3quot is odd and 05 me is even so N tells us at once that J1 I 33 E snxdx E U L wi i39mu39t tllme neecii EDI dcrtai ed iVnt Egratim5 b paIt5 The f 2I1IDWiJ I391g simple th eurem ufmri es the signi cancB of these ideas fur the study Hf F011 rier I5E IT iES Ihearem Leiquot f39c be an Emmgrubkie furmioan de ned M1 the inrervaf w x 2 Ii Jffr even than EH Faurfeer s r as has my casing pRX czind rhg crle cfenrs are giJ H rai 2 V Jr f x39 mam O U X p U Admi if 1f is add rhern its Fbaur Er erEE5 0 N an y sine mrnmv Aand the m ic 39enHr5 are given 0 U mi U IQ 1quotI5in mi bP 53 Tin pmve this we assu4me O that is EuE i Than x mr5 m is FOURIER SERIES AND ORTHOGONAL FUNCTIONS even even times even and by 3 we have 1 If 2 J a I fxcosnxdx J fxcosnxdx 717 7 JT 0 On the other hand f x sin nx is odd even times odd so 4 tells us that 1 1 b fxsinnxdx 0 which completes the argument for S It is easy to establish 6 by similar reasoning Examples 1 a First we brie y consider the function f x x on the interval 1 lt x S 71 Since this is an odd function its Fourier series is automatically a sine series and therefore it is not necessary to bother calculating the cosine coef cients We found in Section 33 that the Fourier series IS 23 sin 2 sin 3x 0s 7 x 2sinx and we know that this expansion is valid only on the open interval 1 lt x lt Jr and not at the endpoints x irr because any series of sines converges to zero at these points b Next we consider the function f x x on the interval 71 5 x S It Fig 44 Since this is an even function its Fourier series reduces to a cosine series and by S we have 1 If a lxlcosnxdx 75 n 2 J J xcosnxdx 75 0 l 2 2 i FIGURE 44 0 e 1FFeseer1sLeeeisrness It is eesy tie see that en set and fee in Pn6 1 en integte tien by perts gitres tiquot H31 P E 5 t tells us that 4 HM E U QI I11 5 Se we have the eitpensiei tree 31 5 i V E El 5 5 5 9 it P 6 0 4 ECDS T The periedie extension ef the initially given fuaznettien is sheiwn in Fig 45 We see at enee fquotI397I the ideas ef Seetieri that ithe series in B eeiwergesst tee this estezesien fer alt I end therefire the E39KpaneiUi39l J is valid en the eliesed rintterrirsl es 5 I 0 ire Sinee 2 it fer J E U the tee series 7 sect 8 are built expenitsiieins ef the same funietiein fx1 1 en the tintenreiii D 5 it Q 0 The I series T is eaiied the Fee7riier sine iseries f lquot1t end 8 is es ied the Feeirier EHEEHE series fer Jr Simiilariy any funetiiieni ft de neid en the intenrei u J E JT that satis es the Dit39iehiet eenditiens there can be texpsntdtetli in th e sine series an e E5 391iI1E series en this intertrsi weiitti the pt39e39 vise that the sine series esrmet E tf Ig te fr st the Ee tjp i i I 0 end 1 as Lmgliesst f1f has the value 0 at these points Te ebtsin the sine series fer ft we rede ne the fuineitiietn if nteiteesseiry tee have tile value if at 3 0 and then we extend it eves the intents emit S I ti 0 is such e was that the BJEtE dEd fuinetien is ed That we de ne x for set E I is U by putitinig ex ax The esteneci ftuineitiiten 0 eltesrly eecli s its Feurier series eenxtsins sienei terms eniy and its eieef ieiein ts are given by 6 5ii39tTIiiWa l ITy we ehtein the teesine seiries fer fist by eisttemzting f tee be see even funetiert en the inteiwel 1 E is If end using b v eeieuiate the ee ieietntsi With respreteti te the sine and eetssixne series diesetrifbed Lhere ewe emphasise peritiietulisrly that the 013E39i na 39f t1nCtiCt i1I is net sssernted in advance tee be case er even ear pte1ietliei er de ned elsewhere at all it is intended ten be en E5fSiEvTI39Ei ll39 45 FOURIER SERIES AND ORTHOGONAL FUNCTIONS 269 arbitrary function on the interval 0 5 x lt r within the very weak restrictions imposed by the Dirichlet conditions Example 2 Find the sine series and also the cosine series for theifunction fx cosx 0 lt x 5 Jr For the sine series 6 gives a O and b q cosxsinnxdx 7 o For n 1 we have b O and for n gt 1 a short calculation yields bn 1 1121 We therefore have 8n b2 3 0 and b2 39 so the sine series is 8 nsin2nx cosx Oltxlt7t Jt4n21 To obtain the cosine series we observe that 5 gives b O and 1 forn1 0 forns l Therefore the cosine series for cosx is simply cos 1 just as we would have expected This conclusion also follows directly from the equation cosx cos x because our work in Section 33 shows that any nite trigonometric series the right side is automatically the Fourier series of its sum the left side 2 lquot a cosxsmnxdx 0 PROBLEMS quot 39139 1 Determine whether each of the following functions is even odd or neither 1 x x5 sin x x2 sin 21 e sinx3 sin x2 cos x x3 x x2 x3 logl x 2 Show that any function f x de ned on a symmetrically placed interval can be written as the sum of an even function and an odd function Hint quotTx fx fx fx f 201 3 Prove properties 3 and 4 analytically by making the substitution x t in the part of the integral from a to O and using the de nitions 1 and 2 4 Show that the sine series of the constant function f x 7174 is sin 32 sin 5x 3 5 What sum is obtained by putting x 32 What is the cosine series of this function I 0lt xltJ r Jr smx 4 FOURIER saunas AND ORTHOGONAL FUNCTIONS 271 11 a Show that the cosine series for x3 is 21i cos Zn 1x 3 no 3 611 1quot 9S53 x 4 nzi Jr 2n 1 39 O S x 5 Jr 39 b Use the series in a to obtain in this order the sums I 1 Jr 1 d 2n 1 6 an Zn 9 O 12 a Show that the cosine series for x is Jr Jrznz 6 x 82 1quot 7 cosnx 5 n Jr s x 5 Jr b Use the series in a to obtain again the second sum in Problem 11b 13 a If ex is not an integer show that coscrx 2 sin cm 20 sin Wt cos nx 2 UH 2 out Jr oz n for Jr 5 x 5 Jr b Use the series in a to obtain the formula 1 a2n2 1 on Jrcotaar Zaz 0 1 This is called Euler s partial fractions expansion of the cotangent c Rewrite the expansion in b in the form 2t n2 Jr co J1cotJrt Jrt 2 t2 and by integrating term by term from t O to t x 0 lt x lt 1 obtain sin Jtx x2 1 s2wg1pl O1 sinJrx x2 x2 x2 Jrx1 iE1 1f53 If x is replaced by x Jr this in nite product takes the equivalent form 2 2 2 S11t1 1x31 2 x Jr 4Jr 9 which is called Euler s in nite product for the sine Observe that this formula displays the nonzero roots x in d2Jr i3Jr of the transcendental equation sinx O d E31FFEREN TML EQuaT1m4s Id The IfunatiIm5 sinix and culsigx am hath Evan Shaw bfiE jft wzith ut valcuIa ifmVn that this idr nt39itie5 T 1 5nr 1 cos 1 7 p Z 1 cos 2 if iil I and A Y EGGS tn Y Y n5 1r EU nrzns T are the Fm1rier s eri35 expan5inn5 Uf thmse funutim15 P Z Frindi the sme series if aha functions in P mblem 14 and verify that Ehzase 4P L 3expan5in4n5 5atis39f3r the identiw sin31 ltr s Equotm39 S 1 16 Prmre thc I rign metim id ent39iti es 3 1 g P I ainjx 2 esin r Sm 31 and c LE 1 r sa153 r 4 0xJ 4 14 and shAw brie y witlrmutV arlcu1amiamnq that them are Elm F urier series expanmnns of the f39unctim5 an the laft A D Th standard form nf a Fmgirier Setri s Q IE1 QTLE WE have wm39lLed with in me prEcEdmg Sectinns where tli1E function under c nnsidcrati0m is de ne d an the in4tE4ru aE 11quot b x J39E In many appslicaIiaDns it i3dESil39 1 ME mm adapt ha 39f1fII391 Bf a F 39 iI iB39I39E s rias to H fungt c n f r e mad mu an iAr1IEr val L E 1 ca L where L P a Apnssintive nLm1her ii erent frm 51 l Jhis 3 done P a change f variabI a that FlquotMZ1lLIl S to a chang mi scams DI the h0 rizmn4tal axi5 q intrUdLmcE ax T E W39 vatriablE I that runs fmm 21 la is as 1 rung from in L k is easy tn reme mher as a statemEnt abrmt rcnpar ti D115 3 Jr LE 3 L 3 L and 1 1 3939Ian INII SCSI The f uncti cm fr is thereiby transfoImamp d inm a fuVn ctic1n of f I 5 E g ff aI1d if we assume that frlt satis as the DirichlEt conditions then so does x uraan tiherefme expaand gm in 3 F uriEr 5IE1f39iES Eff the 39usuai fmni 0Q an 1305 PM P sin mt FOURIER SERIES AND ORTHOGONAL FUNCTIONS Wq where we use the familiar formulas for the coefficients 1 1 quot a gt cos nt dt and b gt sin nt dt 3 Jr Jr Having found the expansion 2 we now use 1 to transform this back into a solution of our original problem namely to nd an expansion of fx on the interval L lt x lt L 4 1 39 fx Err 2 an cos b sin Of course we can also transform formulas 3 into integrals with respect to x 1 1 1 L y a 2 LLfxcos1 2 dx and b Lfx sin1Jfdx 5 We can use formulas 5 directly if we wish to do so but changing the variable to t usually makes the work easier because it simpli es the calculations Example Expand fx in a Fourier series on the interval 2 S x lt 2 if fx 0f0r 251 lt0andfx 1for0ltx lt2 Here we introduce t by writing t a Jrx 2 a so an x X 1 d It 2 2 7 Then gt 0 for 117 lt I lt 0 and gt 1 for 0 S t lt 71 and we have 1 U I anI Odt1dt1 775 2 n 1 a f cosrztdI0 n21 1 1 b J39 sin ntdt 1 1quot Jr mr The last of these formulas tells us that 2 b2 0 and b3 Z We therefore have i 1 2 0 sin2n 1 gt2J1 2 2n 1 39 FOURIER seams AND ORTHOGONAL FUNCTIONS 275 of the origin if the value of this function is de ned to be 1 at x O x wBn B21 er1 U71x BBxix The numbers B de ned in this way are called Bernoulli numbers and play an important role in the theory of in nite series Evidently B0 1 a By writing 1 g 2 B e 1 x xequot1 1 x xe l1 and noticing that the second term on the right is an even function conclude that B and B 0 if n is odd and gt1 b By writing in the form B0 39 32i39t 1 x quot ill 2 3 Bquot and multiplying the two power series on the left conclude by examining the coef cient of xquotquot that 3B IB 332 0 n 5 1B 0 ltgt for n 2 2 where is the binomial coefficient nkn k c By taking n 3 5 7 9 11 in show that B239 39 B4quot 16 Bszzili B2cquot 16 Bl3939g56 From the recursive mode of calculation all the Bernoulli numbers can be considered as known even though considerable labor may be required to make any particular one of them visibly present and all of them are rational 9 The Bernoulli polynomials Bx Bx B2x are de ned by the resulting coef cients in the following product of two power series see the preceding problem 391 0 r r n 71 O For instance it can be proved that the power series expansion of tanx is Do 22 22m 1 B tanx 2 1nl 2n39 2nx2n I See Appendix Al8 in the Simmons book mentioned in footnote 4 ffi nnIFF39EREHTAL EAUUATMIDHS Ia Sthew theft Butt ie 3 39t139t1erntTet mtquot degree at 39that gitfen by the fe tmute pB I v E pE P UT II p rfhquotJ Sherw that 330 EH flit I H U and by using quot quotquot in the pteeedIing preublem st1ew that eTleu B1 Bi flat 1 2 e Sihe er telnet Bx 2 n 1B etl and deduee fr m l39hie that em 1 n t 1 H and H i I 2 Pq H Shewt that A T t B I I39 A 11quot B It I 13 3 I39D Staw that the e0einte series fer the Berneutli peirnt0mie t Bg lfx en the inttereeil U l 1 I is t2I392m 0I tees Zkttet 6 Use the etpaneiet in Problem 1 tetshew that lntxj t 1J T 39139El 2 ztzpyt 5 39 where P 1 e peeimve Enttege r Use the reeultet of Problem m tee Ut39ttaitt the epeeittl eume C ff p dt g te pr 1 2 3 47 P P 1 394 1 2 F t VV M w E r 3 1 b 0 1 e 1 These dteemreriee are alt due to Euietr QT 41 W E 555 s mere sintzfemtettizetn en the btaetaground etf these tormufles see the amete by Rty mend jteuh Euieir End the Zeta IF39unetten Amer ea Met39henzetfeet t enttztyt vent St I IFM pp t AT t 3 FOURIER SERIES AND ORTHOGONAL FUNCTIONS 277 37 ORTHOGONAL FUNCTIONS A sequence of functions 6x n 1 2 3 is said to be orthogonal on the interval ab 2 if 0 for m n 6m 6 d l X X x 0 formn 1 For example the sequence 6x sin x 92x sin 2x 6xy sin nx is orthogonal on 0Jr because I 6x6x dx I sin mx sin nxldx O 0 n 0 for m n I cos m nx cos m nx dx Jr 0 3 for m n 2 We pointed out in Section 33 that the sequence 1 cos x sin x cos 2x sin 2x 2 is orthogonal on JrJr but it is not orthogonal on 0r because 1sinxdx 2 9amp0 0 In the preceding sections of this chapter the trigonometric sequence 2 was used for the formation of Fourier series During the nineteenth and early twentieth centuries many mathematicians and physicists be came aware that one can form series similar to Fourier series by using any orthogonal sequence of functions These generalized Fourier series turned out to be indispensable tools in many branches of mathematical physics especially in quantum mechanics They are also of central importance in several major areas of twentieth century mathematics in connection with such topics as function spaces and theories of integration 13 The formula for the generalized Fourier coef cients is particularly simple if the integral 1 has the value 1 for m n In this case the functions 6x are said to be normalized and 9x is called an 2 As usual this notation designates the closed interval a S x S b 393 See for example the excellent book by B la SzNagy Introduction to Real Functions and Orthogonal Expansions Oxford University Press 1965 50 2T3 DliFFiEEiEN TiiALL lE U ri IHE r h 0inDr m iE 5 equmice in the other hand V I Bx3dA39 1 a in 1 it is easy in 566 tihat lh funmiii cm5 W 94 quotI3939IJ 7 Va are urthoimrmal that is F Mr dx D W W 1 far m n For EX Tmp iE rsimn G P J 1 six 39 25 J EtI5E 11quot J sir nxiidx 3395 T rr 391 fm 1 E 1 1116 mrtl10nmmiail sequence mrrespmiidiinig t the rthkogmnal SVEMLHEIEHAEE A2 is was 5 sin 2 ms sin N P 0K 5 0d Eat at he an rtrthinnmmaii siequienice nf functiiu nisi on r1ib and SU7J39pGEE that we are tryiiing to rexpand 1391JIfh lt ifunictiiinni x in a series 0f the fcrrm mi ii aim p J 6 To deter39mi ne the ieweisicints El we miuIitipiiy b Dj t39h S i dES 0f 6 by ix p gives rxmxii ai 1cxJ ixi Hi uirIiI 0 H 0 M m wh amprt the terims mt WI iI39T coiitain pmc1fuct5 x x with m 2 PL If we aissumta that terimibyiterm iiritegraitifnn if quot3 is valid than by cariryiing D I1t this integra ti in and ii5iI1g 3 we nd that mtSt of the terms diaapipEar and ail thaiti remaiins is fbrtxmicxn ms E J39 ni 39r dr 2 p J I11 deiivingf0Im1Jia 8 fm the iiiei imieiiiti5 in the expaansinn 6 we mJadE two verfy liairge ig mlPlti Fiirsat wig assiiimied that the fuiinctiimi f 1 Gail be iriepre5 intampd by a 15EJiES mf this farm p Secmidg we a umedi FOURIER SERIES AND ORTHOGONAL FUNCTIONS 279 that the termbyterm integration of the series 7 is permissible Unfortunately we have no reason whatever apart from wishful thinllting for believing that either assumption is legitimate To express this somewhat differently we have no guarantee at all that the series 6 with coef cients de ned by 8 will even converge let alone converge to the function f x Nevertheless the numbers 8 are called the Fourier coe icients of f x with respect to the orthonormal sequence cpx and the resulting series 6 is called the Fourier series of f x with respect to qbx M When these ideas are applied to the orthonormal sequence 5 they yield the ordinary Fourier series as described in the preceding sections see Problem 2 below We also point out as we did in Section 33 that the termbyterm integration of 7 that leads to 8 is legal if the functions are continuous and the series is uniformly convergent However in the next section formula 8 will be obtained in an entirely different manner having nothing to do with uniform convergence It will then be clear that there is no need to feel uneasy because formula 8 seems to have been derived by faulty reasoning The truth is that we can use whatever reasoning we please as motivation for the definitions of the Fourier coefficients and Fourier series and we then turn to the problem of discovering conditions under which the Fourier series 6 is a valid expansion of the function f 16 Most orthogonal sequences of functions are obtained by solving differential equations as suggested in the following example A broader discussion of this topic is given in Section 43 Example Use the differential equation y Ay 0 or equivalently y ally to show that the trigonometric sequence 2 is orthogonal on lIwv Let m and n be positive integers If y sinmx or cos mx and y sin nx or cos rzx then H yj m2y and y n2y If the rst equation is multiplied by y the second by y and the resulting equations are subtracted the result is ynyii ymrii n2 m2yy We now notice that the left side of this is the derivative of y y39 y y 39 so 14 Some writers make consistent use of the terms generalized Fourier coef eients and generalized Fourier series We prefer to simplify the terminology by omitting the adjective generalized and to rely on the context to tell us whether we are dealing with generalized or ordinary Fourier series FOURIER SERIES AND ORTHOGONAL FUNCTIONS 281 This norm in turn gives rise to the concept of the distance between any two points in the space or equivalently the distance between the tips of any two vectors dAB HA Bl 13 As our nal bit of review we recall that if u uz quot3 are any three mutually orthogonal unit vectors then every vector V can be expressed in the form V aqlll CYZUZ CY3u3 where a a2 a3 are constants In order to determine these constant coefficients for a given vector V we form the inner product of both sides of 14 with u where k 1 2 or 3 This yields VJc aluluk CY2l2c 399 a 3 3lk and since the vectors ul uz 113 are mutually orthogonal and have length 1 the sum on the right collapses to a single term Vault ak39 The formula for the coef cients is therefore Wk V Ilk i Equations 14 and 15 should be compared with 6 and 8 because their meanings are very similar In essence the a are the Fourier coef cients of the vector V and 14 is its expansion in a Fourier series In the case of genuine Fourier series we work with functions de ned on an interval ab instead of with vectors We speak of a function space instead of a threedimensional vector space This function space is in nitedimensional in the sense that we need an in nite orthonormal sequence to represent an arbitrary function Life is somewhat more complicated in this in nitedimensional space than it is in the threedimensional space described above First it turns out that only special kinds of orthonormal sequences are capable of representing arbitrary functions And second it is necessary to introduce restric tions that remove the vagueness from the expression arbitrary function and precisely de ne the class of functions that are to be represented by their Fourier series We begin this precise discussion in the next few paragraphs and continue it in the next section The function space we consider is denoted by R and consists of all functions fx that are de ned and Riemann integrable on the interval ab Since the inner product 10 is the sum of products of components and since the values of a function can be thought of as its components it mFFrEeeNTLeL EUlJ TIDH39E is netere Iea de ene the inner predeef 1 g ef me f uI391rCei S in 4 re J f1rxdx we Cleeerly U faE 2 fnS pw w ectm v e q lief With 1 11 ea our guide we say that fend g are erh egeer1eeF if their im1er predxuete is zere that is pG e 0 This Au preeeis ely vtghee meeeeife ef rertehege neeleity39 qags given in S eexti m Lq jrcxete dx 0 By the dee nieti ne at t he abeginning Defquot this eeetfin en eereth geeeef Sequence in R e stequenee with the prtUp eLrty that each efuntti ne is erthDgDrnei In eeeery ether and new fumctien is ertAheegeneI it1 eiteelrfe Centinuigng the eene y the Harm of e zfunctmn f is dfe ned by m Uber x1EdxV my Lee 50 mm A I5 fH ifsfle A Eeune39teiee eie ee ed a eu feracerieee if V e if39le 0 er eequieveleeneIley if J fx U A null fuectien snee not be i1ilieeemieeeEly zem For eeempie ifx 7 1 en 0 exeet at the Vpeims x E L L but f 1 at these peingtse tihene f is eee mull mnetieen In the present eenteexet it eenveeeieenet to eensidieer a null funetien as being eeseentielly equal to item see that We functions ere eeneiederreed in be equal if rtiheeier diffesrenee is a null mnetien With this ur1dereIenLiieng me nerm has the eimple epregpertiee llfeff M ilfll le il p Um f U 1f nlL1 enly If f P Twe preeretiites them are net Se elmeple ere i feli L l i Hell 19 Hf 31 t liIfquot1 Hell 29 Hfld FOURIER SERIES AND ORTHOGONAL FUNCTIONS 283 The inequality 19 is called the Schwarz inequality By using 16 and 17 it can be written out as follows in the form fg2 S f2 g2 bfxgx dx2 s b fx2 dx b lgx2 dx The inequality 20 is called the Minkowski inequality its writtenout form is b lfx gmr dx 2 S b rmr dx 2 b lgxl dxl2 The integral versions of these inequalities have a formidable appearance and one might think that probably they cannot be established except by the use of complicated reasoning In fact however there exists a simple but ingenious proof of 19 which we ask readers to think through for themselves Problem 3 below and 20 follows quite easily from 19 by an argument that we give here Thus by Schwarz s inequality we have Ilf gllz f ef g ff 2fg ag llfllz 2f 8 llgllz S llfllz 2f8 llgllz S llfllz 2 llfll llgll llgllz llfll g2 and we now obtain 20 by taking square roots By using the concept of the norm of a function we are now able to de ne the distance a39fg between two functionsf and g in R dfg Ill gll Ufx gx2dx 2 21 We also speak of dfg as the distance from f to g or the distance of g from f It is easy to see from 18 and 20 that distance has the following properties dfg 2 0 and clfg 0 if and only iff g df g dgf symmetry dfg S dfh dhg triangle inequality A space of vectors functions or any objects whatever with a distance function possessing these properties is called a metric space With the understanding that functions in R are considered to be equal if they differ by a null function R is a metric space whose structure we continue to investigate in the next section R JiFFE HEHTML EE Ii iJ 39iquotiiiquot1l Ei ON 9Z At the age hf 13 the RUESii i Tl39CiE139 1 l IfTi thE mEiC7i n Hermainin Miniflmwski lI8 dmw gjiv iwen the Grarid Prize ef the Aieedeimy ef Seienees in iieris fer his lzzr llieint rteeeerehi en quedretsie tTerme etertiing frem e prieiljfliema eheut the rep1rei5en39Eeii iieh ef eh initeg er es the Sum efi ve equaire5a This were ia te1 led tea me eiriieiemm elf e whele new hirerieh of mimIber theergy nnew called the Geie39meh1r enf Lwiwmihere whieh in turn is based en hie highfy eir igiriei quotideas eheut the preperties ef eerwex bfediiee in ni dimieneier1ieii 539peeei In this eenn eeti en he izniI r39eedueed the ehhrieet eeneept ef dietarnee enei3eed the enemies eff eeiumie end 39EU Ff39 acileI and ieemehliehed the imp rte39nti inequeiiisty their beere hie hemei in the years I Minieewski beeeme the meIthemet39ie39ien eef ireeti39it y geeime39t39ri feing me new ei1h jeeit He ereeteid the eeneepit p frJLnr diimeneiene l Sp E39EI i 17E es the preper meihiemeiiieaI setting fer Einsteiinie ieie5enieielJj4 phyeieeii end minmeth emetieaiji whey ef thinking eheut 5peeiei re39ietiwi39v in re new femieu5 ie eturrei ef 1903 he hegen with e eentenee that is mete eeeiflgr feegettiein Efrem new zn spaee by iiteiel emd tirhe by itselifi ere dieemeI hI THHE awiey inm mere ehediJwe amt einhr a kind eel Lmien ef the we will retain an indiepemdieini eerietehe ef X Hermeinn Amecifeue Sehwere EiiH43 iQ21i 339 ipupil if wejigr5tra55 wham he 5L1eIeedeei EEFijiI 39l made euh5ten tiel e nhquotihut39ieIiin5 he the IhEti39vryief miniiI1el eurfeees in gieemeit39ry am we heen fermiai imepiing fpeatentrieii thee ry hiper geerrie triiie fuiinetiens enel ether memes in enalreiisi In C z f fil i i mappingr he rescued and rigereu5lLy39 neiied deem eeime en Riemenn e 39 re393quot impenent but iremher inIuitive dcieemrerriee eefp eeiaiv39 Ihe hasie iemerm rhepping1he I39em In mimmel eurfeiees he gave the first IIquot39igDIquot rl39iLS preef ehei e sphere hee re emmleir eu eee area then Fiquot ether hedy 0f the eemse vehlm He aiee diieeewered end mewed the peeie 39lrie39wgiequot eheeirem wf eiemeinIeey gee meiny In any EilTEi ifigi d imiengiie the ineerihed Lrieng ie wiiIh emeiieet i39IE1iIquotiii11E3iiIEi39i the ene wheee 1rertieee ere the th Tee feet cjifi the 1ltih1deeif the giieein trieriglequot5 PROBLEMS S Dine ef the imp fi l l II lC39imquotJ I qfIliiE39 iII 1EE ef the erth egeneiiity pI39EpEilquott39iESi ef the 39trigenemetriiie eeqeeinee 2 ii EiiimEi39 e q39uetiene 4 in this seerlrieh and 2 5Ju iii 3 in SEE39I 5 iDrII1 33 is Bee5eiquot 39si i EsqH HI If fax ie emr f1Jr39i tjiD I39i in tiegreihie en 2Ie its erquotdirie r 39 F es39urier eee ieieh39Le eatiefyr the inequeIitF i ii 2 in man 2 J R H k H R Pirmre hie by the fie llienwing etepe 1e Fm any 1 2 1 eie ne 1 5 Earn i P mt eeis he hi sin ii t 739 Jae Fer drteii5 eee Che ep Le39r 5 ell T5h1demeehier en d Tuepiiiz The E39ve39eymeri ef Mearhemnriee Wine39eteni Llniweeeity Pr39ees Ii 39J3939JT er Ceerem end Rehhinez Wher hr fPir 39J39i39hE fTF fi 39fT Dei39e nrd iU i 39uquotEiquotSiIl39 Freeeni I93941L ppiu 34i 5i FOURIER SERIES AND ORTHOGONAL FUNCTIONS 285 and show that 1 H 1 2 n 2 2 quot fxSnx dx quot 5510 Z ah i bk39 T5 1 kl b By considering all possible products in the multiplication of sx by itself show that 1 H 1 fl sltxgtVdx a 2 at bi Jr Jr 2 kl c By writing 3 It mx sxFdx 75 J 1r fx2dx 1Jfxsxdx J sx2dx J 1rJfx2dx 1a C21 ai bi conclude that 1 quot 1 quot 2af 2 a bi lt I fx2dx kl Jr 11 and from this complete the proof Observe that the convergence of the series on the left side of implies the following corollary of Bessel s inequality If a and b are the ordinary Fourier coef cients of f x then a 9 O and b gt 0 as n gt 00 2 In the case of the orthonormal sequence 5 verify in detail that the Fourier coef cients 8 are slightly different from the ordinary Fourier coei cients but that the Fourier series 6 is exactly the same as the ordinary Fourier series 3 Prove the Schwarz inequality 19 Hint If g at 0 then the function Fa f crg2 is a second degree polynomial in a that has no negative values examine the discriminant 4 A wellknown theorem of elementary geometry states that the sum of the squares of the sides of a parallelogram equals the sum of the squares of its diagonals Prove that this socalled parallelogram law is true for the norm in R 2 llfllz 2 llgllz Hf all llf gl2 5 Prove the Pythagorean theorem and its converse in R f is orthogonal to g if and Only if llf EH2 llfllz l8ll2 6 Show that a null function is zero at each point of continuity so that a continuous null function is identically zero 38 THE MEAN CONVERGENCE OF FOURIER SERIES Consider a function f x and a sequence of functions px all de ned and integrable on the interval ab There are different ways in which 0 T3tiFFEilEHTiAi Eeu eTe1ss WEligti39 D eisn eetweitge te ffxt anti these site hest unders teedJ Ut tem1s ef the ptquotehtleh1 tel eipipttesitmetihtg is by pitxia If we try to ieppireisitmste f t he ps then each ef the mimhersst fisii pnlisstii and ifixii ssisiiz iii gives a 1tessure left the ermr in the Eppr Jt im I iDi1 at the jpexint s It is elesr that if we est these m1mhets is smell then so is the ether The usttei tie nitiien ef eenviergienee snteiuhstst ten the s39t temB1 Itt tha1 the S Eq1 L1E1 C ef ft1rietiet1s w x eeunvetigs the the ftme tien if P m ejeeht paint I either ef the E3fPIi E55i Ii39i5 1 eppr eseTh es setre es rt rt r This is the sfemtiiisr eieneep t used in SsEt3iZiampD39I39i5 33 te l and for ehvieus rieisisens it is eslleri pieimwise E rRiJErgErmEE On the ether hemit we might prefer te use e meesttret ef eitherquot that refietrs te the whale interveI i simu lteintteehsiyV ihsteeel ef paint he peifrit We eat ehtein sueht he miessure hy integrating the EltlJIEp nESSiDiquotiS 1 free at test a 1 P E fcxt mrale and Lexi The seeend integzrei here is heitter eheiee than the First fer twee l39EH539U39 3l it s391lZiiiiCiE the sewtrwatd ishsehite vrshse sign in the rsts tintiegtei end the espenetnt39t smashes Ittenty ef the necessary C 1t3MialiD1Ez very eehvenient te cherry eut es we will see b quot w i The measure ef Efr E we adept R t i39Iei39E fDf E3 M V time s as m T39his q39UHtntiIjf is esiieti the m39 Et3 sqtmre errer The tertmizmeiegy his sfpprepxrriete bEt 39t15E if the intetaii 2 is P by G i e the t39estuit is etsteteitir the mean ssiu e ef the squerte eirrer fti p x2i If 2 eppresehtes sets as is K es the seqtiehee pmsi is J ten eetrmerge in the mreen the 1 end this E i pii D eslied steers teteeuterteeeet Wet semetimes symheiize this mitimie est eenweirgeheie hy Wt l7 iI39i39ng ffxi whrere iim stehtis fer in the miEEl Our d iseussieh w7 the test tirf this seetien will shew Tthst in the ease ref Fetitier series mean ED u 39ETgEi CE is imueh eesisetquot te W39DIquotk with then erciinier1 1eintwis e eens39etrgeneet We sssumevd at the hegintning that the fiutnetiens fi x ends pntsi heieng tn the ftinetien speee 4 described in the pte eediitng seetyient We new peittt eiutt that the messes squtstte errer 21 is gprieeiseiity the square ef FOURIER SERIES AND ORTHOGONAL FUNCTIONS 287 the norm off p in R 5 b mo pxgt12dx Hf pn 239 3 The mean convergence of px to f x is therefore completely equiv alent to the convergence of the sequence p to the limitfin the metric space R namely dfP llf Pnll 9 0 as N gt As indicated here we will often use f and p as abbreviations for f x and L px in order to simplify the notation We now come to the main business of this section Let x be an orthonormal sequence of integrable functions on ab so that O form n ltzgtmxgt nlt xgtdx 4 1 form n We consider the rst n of these functions xgt xgt2xgt xgt lt5 and we seek to approximate a given integrable function f x by a linear combination of the functions 5 Pnx b1 1x l b2472x quot39 bn nx Our purpose is to minimize the mean square error 2 En J4 Z pnlzdx J4 i bl l 39 39 39 bn n2 dxi K by making a suitable choice of the coef cients b b Our rst step is to expand the term in brackets in 6 which yields 5 b En f fzdx 2 wt bmfdx b 439 171491 quot39 bn n2 dx 7 If the Fourier coef cients of f with respect to the orthonormal sequence qbk are denoted by b ak J f k dx as in Section 37 then the second integral in 7 is b J b1 bcpfdx a1b1 ab 0 mhF39FEfREHTlhAL EeD hU TUNS The ihimrd i tegraii in T can but wrmhemh E J 39 B a E quotE g br39ar hrI xh J Pu p 1 z i 1 H a bi where the semhnhd gharup of terms r h quot E IEi S p11 Illdu ts 51I E if and the hal val uE results fmm ms1ing 3i Lh Tiheaa mnsidewatmhh5 enablm L15 E0 w1ritE the m square Ermr T as I I Vivi x I Fair s u kbk h t t my gr H1 5 It I if we haw nhtice that 2 ebt P u H bx quot 39 2 lhh ng the fmmula fur E takes its h hnahl farm P nr 1quot E J air Hr mi g PM amp am 9 miquot kill J Al E rmuia 9 fm the mean sqquareh errfnr En has a hnruhrhfber uhf ihmpmrIam cxnseqhuhences mhhat fuhnw by very Sihifnpl ha5 hhiinh Fhirsh th hlerhms bk 392 d azre psiti ve N1E EE39 bk mg in which has6 they are z rD ThE39fBfD39T E chh uice hf the by that minihn1izhes E is Db viUuE 39 bk 2 Hm and we have Threnurem 1 Far Emh phh5fhrr39uE39 integer me39 mi pmfaihi39 mm hf he Fhurier herfeh f hrmmhfy El P t k 39 p 39539 i 7n 39wahn I5 quotI ghwE5 H Ehmhhhr merm rquarE eirmr 3 GV A S ix hhn 393gh 391a n by any c hm rm r HIE3i ff Jhf E pg b E m E 23 Fuhrrher rim m39n mun uv1fuE hf the Ehrrmr 53 1 may Fmrmula g tails L15 hathart Awe hhlways have AEH p II becau5 e the hihlltegmnd in being 3 5qua1r c E35 hhnnegatiwB Sihs E U fur all Eh i ES IE thhe hi it is l aar that the lquot i 39i39m U 1 value of E whi h arhisamps when bk am is also 20 Tharefnrer 10 implies that war pZ hr hi E J fl n39 39 FOUREER SERIES AND ORTHOGONAL FUNCTIONS 289 By letting n gt 00 we at once obtain Theorem 2 If the numbers a fqt dx are the Fourier coe icients off with respect to the orthonormal sequence then the series 2 af converges and satis es Bessel s inequality a s h fx12 dx 11 Since the nth term of a convergent series must approach zero Theorem 2 implies Theorem 3 If the numbers a fZf gt dx are the Fourier coefficients of f with respect to the orthonormal sequence then a gt O as n gt 00 Theorems 2 and 3 are obtained for ordinary Fourier series in Problem 371 Here they are seen to be true for generalized Fourier series with respect to arbitrary orthonormal sequences F For applications it is important to know whether or not the Fourier series of f is a valid expansion of f in the sense of mean convergence This is equivalent to asking whether or not the partial sums of the Fourier series of f converge in the mean to f that is whether or not f lim am 12 n poc kzl In view of Theorem 1 it is evident that we do have a valid expansion off if and only if minE gtO asngt00 and by formula 10 we see that this happens if and only if Parseval s equation holds b 1 ffdx a20 a kl We summarize these observations in the following theorem Theorem 4 The representation off by its Fourier series namely faiQba2 2quotquotl39an nquot39 13 is valid in the sense of mean convergence if and only if Bessel s inequality 11 becomes Parseval s equation j ai h fx2dx lt14 23990 mFFEeEH391tIssnEautJsTttts439s If e Pewter espansiena ef me term 13 is valid in the rsettse set meant seenuetrgieneet fer every tttnetin gt in j then the etttthetntermail sequence m t is tsetd he he quot F fIf m EIE U eempltete sequettee th EtFt is EL s equemet qb that scene he used fer mean squstte ttppl irI1al i nE of the farm 12 fm tertttittreryr tunetiLe ns f in 05Q It est be prsmquotEd that the 39t g 39n 39Tn39EtItE settenee sin 3 ms S sin V N R 9 Q xR is empttete on JIJT39 i 15 Remmk SS The pmof ef the theexrem jttst stated absent the tttigDne mettrite setquenee 15 is lung and wttuld tektet us smtteta tee fat e eLld39 Herweveri if we teesti Ptetzslem 2 in Seetitrrn 3 then we see that this ttte tem immtediet eIr yitetlds the testilewilng tttsjet t tntEJ39L139S iitit I391 whteht Catt be itttwete preted as sweeping sway am the d gtif eut1tiest that arise in the tl1e ery eef petinttwtse eerzwetgettitee ferrt Feuttet seties gt tr39aepresemed by Ev39t tardtnttry FTmrt er series me sertse Hf meme use It tJ E39rgEquotIt Equot Theorem 3A If at its titty fttntftttm ds rted and tquotttte39gmttt 39e39 n W J39T39JTL then 1 T HI k1 an 0j9 sen esnx tsH sin G1 1E1 an r I esters the e tma Era are the erltffn erjy39 F LtquotrE r eese etrquotents tzif f TD appZFEEiaIE the clean sistmplictw2ft1tiststa ttemettt it Lhelps ED teteH fttemt U111quot pntevieus twesrk that this reprseste ttte39tiOn theere m is fsfitset if 16 is t terpl E nEd in the sense eff 39pD39it1IIWiE E EEJl1quotsEirgEnE39E further thee tepttea set1ttst39i tttt ettetnt gfeits tfer some eenttitnttttett1s ftlttvct fl Remseta E In Prehlem 0y belew we ask the student Ate 5hErW that we sptecietliiee to the intsewetl M 139Ts sTFJ end use the D39diFl J339 Ftmtritett ee e ietents then Persevsws equati n A11 tekets thee terms 1 quot p 1 5 st n L E J Lmttllgrirt 3 2 mt 93 1 The functtin ft in this eqLtt1titnn is I1SS39l39IEli trJ beeljtmg ten H that is tee es R l mt inttegretble en tet and fer teeny such futttettern its ssquste The l ssh ttmls ftzit the E hft1ItJIf we J39Htt39Ft in mind are ftwu m jl f theerems Hf elsssrieet El l31l395tEi Ft39 tquotrquots strnnt bItts l hE39t7 iI39 quottL39c39 Jquotti tmtl ttI39tt Wet ertet rst3t tiiT39FI IJ39a39TIJ tIiquotvi39Ti stJ5tttZ r a 1 tJ tttttretrt FOURIER SERIES AND ORTHOGONAL FUNCTIONS 6 f x2 is also automatically integrable It therefore follows from 17 that for this function the Fourier coef cients an al bl a2 92 have the property that the series 2 a3 bi converges Of course we already knew this from Problem 1 in Section 37 However if the Riemann integral is replaced by its more powerful cousin the Lebesgue integral then this statement has a converse that was proved by F Riesz and E Fischer in 1907 The famous Riesz Fischer theorem one of the great achievements of the Lebesgue theory of integration states that given any sequence of numbers a0 a b a2 bz such that the series 2 a3 bi converges there exists a unique squareintegrable function f x with these numbers as its Fourier coefficients It is customary to use the symbol L2 to denote the space of functions fx that are squareintegrable on Jr 71 in the sense of Lebesgue where as usual two functions are considered to be equal if they differ by a null function When Parseval s equation 17 and the Riesz Fischer theorem are taken together we see from this discussion that they give a very simple characterization of the functions in L2 in terms of their Fourier coef cients It is remarkable that no other important class of functions has a characterization of comparable simplicity and completeness a fact that delights the souls of mathematicians NOTE ON PARSEVAL MarcAntoine Parseval des Ch nes 1755 1836 member of an aristocratic French family and ardent royalist poet and amateur mathematician managed to survive the French Revolution with his head still on his shoulders but was imprisoned brie y in 1792 and luckily fled the country when Napoleon ordered his arrest for publishing poetry attacking the regime He published very little mathematics and none of any distinction but this little included in 1799 a rough statement that only slightly resembles Parseval s equation as it is known tp mathematicians today throughout the world and for this his name is immortal PROBLEMS 1 Consider the sequence of functions fx n 1 2 3 de ned on the interval 01 by O 0 S x S 1n fx 1n lt x lt 2n O 2n S x S 1 7 It should be pointed out that L2 contains R and many other functions as well and that whenever the Lebesgue integral is applied to a function in R it yields the same numerical result as the Riemann integral 292 DlFFE HEthfTlth1EDUH39TiE N5 at Sihew that the stetquence r Ztlt rentrtemes pewirmzwitse ten the ZEt fL1I39l39CtttJITI1 en the in39tetve b Shew that the Eeq391J1rEI tiE x elhee netI ee ntve rg e in the mean In the zero futne tiim eh the iY391t Kr a1 MDttt Chneider the tfhfltlewithgh 5eq39ut enee f El dl SEIVmJiNt rVlampm539 hf lttl et t tj 0 P H ht tt1j4 end diettete the 5 5tuettnen e1 hry New de ne Es hequte nCe hf funettehe 39iIt39 em Ut1 by 1 teirx in I quot 39x U 39EU T 1 net in Ce Shem that the eequehee Ut C DHVE39TEE5 in the mean he the zero funetien cm the itnt ervel M Shaw that the sequence tf1 dkuee net entiwerge JEI1il39t t39WihE at any paint at the aintetwet 7IU lL Obtain the fI3I39rI39quot1 llJl bk 2 at frhm hhthh 3 uted 9q Lttihg the feet that 3E39 ef539hi E erfhert Ea has he htihimum vahte 4 The efunetfttien gr t t is to he hepttrtU ximtetetttt en 3939f he Kpl t r h sinr einlr hstin 3 sin 42 begin 5 in tsueh e Away that H 1 pri1 the is H39lfiHiITtiZEd What values shhttld the ene Iete rtIh ha have 392 5 The functtian fx I to he epprtmzintetett en mr hy bu eriht etinltf heath 31 in suteh a way that E pt7Ed1t 15 rhihtimited What treluee reheulel the t E tI39r tiE139TIt397 hhE39i rE39 I p T Shaw that P39ar5ei39e lquotS equetieht 14 hee the ferht 139Tquot when the r39FEh mtIEm l3 eequehee x is the trigenemetrie teqruehee 15 7 Ot3 tai i the sums 39i 1 lP J 5 j t L H 90 by P39pjft1 g Parsetett t5 e tuetien n in the tpre eding pmhilem ter the twe FtJtJr39irer39 ee ee B q Sin p J eitrt 31 N x Elflax 4 M E dI and These settiee are fetthti in EtempLle G eh P39m39btle mt 3 t1 1f 3 USE the tT1etheitt hnttl rttshits hf Pmhtem te hhth39in the stunt oA Eta G G oA fmm the 51ne series fur 3 F rmh39lem C FOURIER seams AND ORTHOGONAL FUNCTIONS 293 9 Use the method and results of Problems 7 and 8 to obtain the sum H 2 H 2189450 from the cosine series for x Problem 3512a APPENDIX A A POINTWISE CONVERGENCE THEOREM We divide the work of stating and proving the theorem into stages for easier comprehension 1 Our rst purpose is to obtain a convenient explicit formula for the difference between a function and the nth partial sum of its Fourier series This formula will enable us to prove pointwise convergence for a large class of functions that includes all the examples given in this chapter To develop this formula we begin by assuming only that f x is an integrable function of period 21 The nth partial sum of its Fourier series is then 1 fl sx Ea I a cos kx bk sin kx 1 1 where 1 quot 1 quot ak ft cos kt dt and bk fts1n kt dt 2 By substituting 2 into 1 we obtain 1 It quotA 1 1 sx ft 2 cos kt cos kx sin kt sin kx dt n k1 1 1 quot J ft 2 cos kt x dt 3 H 41 2 kl If we de ne the Dirichlet kernel by 1 fl Du 4 2 cos ku 4 2 k1 then 3 can be put in the more compact form sltxgt i ftDt x at lt5 294 DIVFFEREHTIAL EQEMTJALQHS P l ii g M 4 I J in 5 i39i hI5 a1x NF A m dun 6 quotv n By thus Li iti n 4 D1u has pEI ii d Zn and as a fLmCti0n Gf 1439 H9 4 alga has period 271 Theref re the integral of fr uD u wet amr intawal Elf length 323 Equals the integra var any other interval nf hitngth 23 and 6 can ha written 5Ix jfr uDHu du T Si DM D39Hu we can rveplarzzt u bjy u in Z 10 bIain 1 H ms awn u M I 11 x F 83 and adding F and 8 yields 0 0 M PA fix u1muJ an The integr4and here is an even functtinn nf M 50 Elm integral from quotquot3 tn at is Iwi the integral fmm U tn n and WE ham s39x ij f1 u x uDu du 9 TE bring r inm ur disc u5sit1I1 and put Ih e diffBIe nce 31r x intm a rmnveni rnt form we ncstice that 1 I Dudu g M gt 2 sine the ter39m5r 05 km in 4 integrata to mr rr If we nmw multipljr miss by 2 we Gbt i fix i 2ftxDm P mm and sub tractir1g 10 frcmm 9 39iE39l SIS xx fgtr 3 Hfix N me 2rltrDm an M This frrrmula is mm fundamampma1 mm for s1udy71r1g the c rwe1rgem uzvf s x i fxIl FOURIER SERIES AND ORTHOGONAL FUNCTIONS 295 2 At this point we need the following closed formula for the Dirichlet kernel 4 sin n u E1 COS ku M 12 Kt lLW if sin u i O 8 This enables us to write 11 in the form 11 sx fx 11 gu sin n u du 13 where fx u rltx u 2rltx 2s1n u 1 gw Of course gu is really a function of both it and x However we are going to be examining gu with x xed and u variable and this notation helps to avoid confusion In view of 13 to prove that sx 9 f x as n gt 00 we must prove that If lim gu sin n u du O 15 n it Our task is to give a rigorous proof of 15 with appropriate understand able and clearly stated assumptions about the behavior of the function fx 3 As a preliminary to the proof of the main convergence theorem stated below we need the following lemma Lemma If gbu is integrable on the interval OJ39r then If mi mmnMn9mmn m I196 0 Proof By the addition formula for the sine this integral can be broken up into J u cos u sin nu du I gbu sin u cos nu du 0 0 B This formula can easily be proved by writing down the identity 2 cosl sin B sin A B sin A B n times with A u 2u 3u nu and B 142 and adding the results to obtain Zsin ucosu cos 214 cos nu sin n u sin u 296 emFFeaE1ariat EauuaTteHs if we smite p s A Mia sin o ems rm du 515 in anti 7 2 H 39C3iI39S n p sin F1 aE Gilt i 1 then the interai 16 is FE Bil 39 2 J it is easy ta see that A is the nth eeIe eienit in the eesiae series fer ii sin in and B is the nth eee ieien39t in the sine saeriies fer es u ees at Sinee tibia is itttegra39bie each ef these f unetiens is alse ilntegraili ei It new ieilaws tram the eerte ary ta Eesset s ineqeuality stated at the end at Prebiem 311 that A s U and B I D as n a 99 and the pree t39 et 16 is eemplete 41 in View eii eeaditien 15 and the lemma all t ia it remains is he fermulate assumiptieins sa eieint ten yguarantese that the fanetien gsi de ned thy 14 is integrabie en ir So far we heatwe mil the general reiquiremenitsi that fx is integrable en i am and periieidie with perieci We new maise the fiuirther assutmptien that fx is piecewise sheath en ii 39Tl a1391 Tltis means that the graph en 21 eensists emf ta init mimber ef ee ntit1 uJus etiwes en each of which fquotJ exitsts and is ee39nJtir1iuevus It aise means that the derivative exists at the emlpei ts ef thesie emwes in the sense of Jim jX H H and lint fcx M fIn Ie wi H u i39tiir II in this way the mnetien fx is gtlarantesed te haves a right wderivatitive and a left dEiTi ti39 tiV39E at ever peint ximignteiutiing paints of dizseeintimuity twhieh we denete fLa and fim Of C39DttI S the fun tiiitm fc is atlilewed to i391a1re a nite iaumber of jurmp iseentinuitties mi i iquotITJEquot3939TJm Hewes39er since the Feurier eeef eients are not changed if fJ is rede ried at a nite number elf fpeints we may assume witheat less elf generatlity that fare 0 few 2 at every psaint t1 whetherfa1 is eeinttimaeus at at er nets Our peintwisie keeavrergteinee iheeirem eaJn new he states as feilewsi FOURIER SERIES AND ORTHOGONAL FUNCUONS 297 Theorem If f x is piecewise smooth on Jrr is periodic with period 211 and is de ned at points of discontinuity by 18 then the Fourier series of f x converges to f x at every point x 5 To prove this theorem let x be any xed point We wish to establish the correctness of 15 and in view of the lemma it suffices to show that the function fx u fx u 2fx 3 quot Zsin at 19 is integrable on 07r It is clear that the only doubt about integrability arises from the fact that sin u 0 when u 0 for elsewhere in the interval sin u is continuous and positive and the numerator of 19 is certainly an integrable function of u on OJr We see from these remarks that gu will be integrable on 0Jr if we can show that gu approaches a nite limit as u gt 0 By using 18 we can write fx u fx u fx fx gu 2 sin u in u rltxgt fx u rx at u u sin ll But as u gt 0 17 tells us that fx u fxgt ax and fx u rltx gt ffm u u and we know that 1 2H I gt 1 smgu It therefore follows that 8a gt fLx f39 x so gu is integrable on OJr and the proof is complete DIFERENTIAL QUAT1 NS IUNDAY PR LEMS 39 HISTGRICAL 6 iEhrory39 iii FUU i39EIquot series dicus5ed in HIE precedingi chiiapiter had its hist riml 0rigin in the middie mf the eigiihteanth E E1tVL1Fr5quoti wh several m IIhI Ia tiEfi S were 39 39liiLJIidf 39i 339ilg the vibiriatii ns of 3tr etchieiri 5triiings The miiithemiaitiica1i thenrry of titiese vwiiiibmiitzmns amUrums 10 the pmhlem Bf solviiig the paritiiiaii di ereiiiia equizimm 8 55 H 812 I3 I 4 ii 1 where 1 is a jp siiiiiim constant This miEdi imeri l9niia i iwmie EqV fri m has many s lutiozns and the pmiialeimiii for a particular viibiriatirigi swing is tu lih 5Diiitimi thaw satiisi eis certain preliminary iC1EZIlitii39El11I5 as5miateci with this is tringii siicii as its initial 5hap e its initial veiUcii3ri Eire The soiutii n 1ihEI 1 de5riiibes tile Subisequieiiil rjnmtilmi if the string as ii iribiaie L1lquot1LiEI iensi in The equiiibriun1 J DSitiiD n f the stritrig is SS7li 39iEd t1 be aiming the I fii i and if y yxi is thin desired 5ol39I1imic3ni ruf 1 than quot5331quot xaid iwiiue DE Iquot 3 I the E1LWquot J f y xr giwas me Shape Of EhE iiiii5plaLErl 5triiri39Ig at that WIATIl39H1EH39 See itht dai h d Eurve in Fig 136 and itihis shzip E lE 1I1gE 5 fmin mifiniiem to imiimianiii c PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS pZ FIGURE 46 For the case of a string stretched between the points x 0 and x Jr and then deformed into an arbitrary shape and released at the moment t 0 Daniel Bernoulli in 1753 gave the solution of 1 as a series of the form ybsinxcosatb2sin2xcos2at 2 It is easy to verify by inspection that a typical term of this series 1 sin nx cos nat is a solution of equation 1 Further every nite sum of such terms is a solution and the series 2 will also be a solution if termbyterm differentiation of the series is justi ed When I O the series 2 reduces to y blsinx b2sin2x m This should give the initial shape of the string that is the curve y yxO into which the string is deformed at the moment t 0 when the string is released and the vibrations begin see the solid curve in Fig 46 However d Alembert in 1747 and Euler in 1748 had already published solutions of the problem which for the case stated above have the form y gm at for am 3 Here the curve y f x is assumed to be the shape of the string at time t 0 also the function f x is assumed to be de ned outside the interval 0Jr by the requirement that it is an odd function of period 21 In Bernoulli s time no mathematicians had any doubt that in nite series of functions can be differentiated freely termbyterm Such doubt was the product of a later more skeptical and more sophisticated age W JiFFEHEtiTiaL EEt39Ua1 1U39i4S that is Max iftxpi and fix p 2 2 is if we eempatre the 1Iil1tiUiT1 ef B fITtDlIiii with that ef d tAIrem herit and Euler them we See at enee 9 e might t have hereause this is what we et iii the selutiiens 3 and 8 area at time I 8 6 Tthetreierie as a reauit ref rnathematieaiIy t1 ijp39EijHg this phy sieal pr hiemrt Berrieu i wriiivredr at an idea that heat haci VEfjf farreatething iI39i lJE EE er the iltisterg ef matihremattiee and phyeireai eetiernee I1amseiy the pasithtitlity that a funetiegri as enerai as the ehape elf art rarhittrariiy deifmme taut string cart he EIiEpamp1 dEd iri a iri DmE tI iE eeriee the 4 Beth diheiermhert and Enter rejected Berneliiquot a ideat and fer eeeeritialiy the earr1e reaeerrt it is clear en physitrerai gereunde that there ie a great arrleunt ef freedom in the wat r the string ears he eerrstrair1 edt in its itrritiail prraitiierrr For example the eIritng is plucked aside at rat singlet petiqrntie three the shape wiifll he a hreherr liner 4a arid PJM is puehedf aside having a eitreuiar ehjeet ef S m kind then the shape will he paritiy a straight line partly an are ef a eijrele and p Iquoti5I ane t her straight iine ae in the it rweaseriahie te expect that the aingite ferrnuiaquot er ariaIrtie e eprerasietniquot 4 eeuld treprestrernt a stratigthitt line en part at the inttretfvai 0 ta eireie en atnether piarrt and a e eeerId atraithtt line mi etiil anethrerr part Te the rrIuathematieitane ef that time eiznreept Berheuiiit 0e seemed ahsurCL Te duhiermherrt the curve in Y weutld i39l I39tquot39E 139Ep1 1SE39nIEEi three separate rraipha eff three distinct Vfuin tiiv ii gi rnergteijy pieeeel tegethert Te Euler it weruld have heen a Si zgi graph hut ei three funetiene raither than a aitngier fl1ICti TIt Beth riitarrltiseedt the jpeeaibiltity that euieh a graph eeeidr be I39Ejp IquotE5E tFEiCii 0 a single rreaarrrtaihie Til1 vEti t iihe the series 4L The eeintrevere3r bubbled en fer rruahy yiearia hand in the ahsenee ef rnatherrnatireait ipreefe he erte eerrrertedi arrmne eiae he his way ef Iiquotii li39i39I1g The rrmre general term ef a triigenemetrire ser iea CUI139iEi ifI lgi heth U T T 1 ti r T T n n fiui mm W 39ir PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS 6 sines and cosines namely 1 fx Ea Z a cos nx bs1n nx 5 nl arises naturally in another physical problem that of the conduction of heat In 1807 the French physicist mathematician Fourier announced in this connection that an arbitrary function f x can be represented in the form 5 with coefficients given by the formulas 1 II 1 J a i fx cos nx dx and b fx sin nx dx 6 No one believed him and for the next 15 years he labored at the task of accumulating empirical evidence to support his assertion The results were presented in his classic treatise Th orie Analytique de la Chaleur 1822 He supplied no proofs but instead heaped up the evidence of many solved problems and many convincing speci c expansions so many indeed that the mathematicians of the time began to spend more effort on proving rather than disproving his conjecture The rst major result of this shift in the winds of opinion was the classical paper of Dirichlet in 1829 in which he proved with full mathematical rigor that the series 5 actually does converge to the function f x for all continuous functions whose graphs consist of a nite number of increas ing or decreasing pieces in particular for the functions illustrated in Fig 47 Thus were Bernoulli and Fourier vindicated We must add however that Euler found formulas 6 in 1777 but believed them to be valid only in the case of functions f x already known to be represented in the form 5 As we know from Chapter 6 in recognition of Fourier s pioneer ing tenacity a trigonometric series of the form 5 is called a Fourier series if its coef cients are calculated by formulas 6 from some given integrable function f x Those readers who would like a more detailed description of these memorable events in our intellectual history are urged to consult any or all of the following masterly accounts Philip J Davis and Reuben Hersh The Mathematical Experience Houghton Mifflin Co Boston 1982 pp 255270 B la SzNagy Introduction to Real Functions and Orthogonal Expansions Oxford University Press 1965 pp 375380 and particularly Bernhard Riemann in A Source Book In Classical Analysis ed Garrett Birkhoff Harvard University Press 1973 pp 1621 In the next section and its problems we present an organized exposition of the theory of the vibrating string sketched above and in the sections after that we turn to other applications of Fourier series in physics and mathematics l3I39FFEREH 7HAL EDLJe quot riEiH5 N P eleen Ie Rand d39 39Mem beIt 1T1T 1hE3 ems e F Iemeih physieEet metheemeetieeihen erzmd men ef ilenere In eeienee he is reemhembered fie hd39eh39ieembheerrquote principle in mechanics end hie eemlriuzzun ef the weee equetien The men werk ef hie Me was hie ee aiberetien with hDieer39et in prepereieeneg the Imettefe femeue ErhyeFeped he whie h pl ereel a mejer rele in the French Enlightenment by emphasizing sciehee eI1d literet11r39e and etteE39King the fereee ef ereeamien in ehureh and etate Mem39eert Wes Fl eehled friend ef E39ueEe1 Lagrange and LeepEe ee 40 THE VIIRATIMG 1 We beh1 by seeking a ne1ntrhr39ieE eehlutien yfx ef the Eql iI5I ZiIeIT1 y 39 lye 0 1 that satis es ebeieehdary eeleiithms MU U and yn E D 2 The pagremeteer is quotfree tee aesume ahyr reel meme wheteever and pearl ef ever ask 0I tea digseever the 535 her whieh the pmbEm term he ee1ved In em previees week we eheme ceeneidered ehly fnhEe 1m39ue praebfem5ee in whieh the seieutiezn ef ae eeeend erdenr eqquotuetiehe ii seught th at satis es twe eendimie nes at e eingleee value ef the i dEpf39e dEnt veriehlehe Here we have an entiereeihr dei ierenyt Si39E 3TZei A39 fer we wish he satisfy ene CD ditiDe Je eeeh ef twe dieteinect values ef x Prefbeleeme ef this fleeined are ceheed beueedery tmfue prebEemi and in general 39lE are mere di ieu u t and farreeehinVg in berth theeeey end preetire e then irlitieelg valhue pmbEe m5 In the pmhrm peeed by 1 end hewieveree ethere are he di ieuletieei If 1 is hegath e then Theerem 0C telle Us that enly the terivieel seutien of 1 can Ev tii fj 2 end if C M t11en the general 5 lU tiE JI1 f 1 is ye eJ 4 and we hev39e the eemee ee neh1eien We are thee r39E5II tied In the ease in wehieh A is peeeitive where the egeenaexrefl E D lU ti1 ef 1 ie y xh 1 sin P 2 3 EDS I and since JM I meet he 0 this re39dueee ten C sin T39heusi if ram pr emlem has 3 eeILMien it must be ef the farm 3 Fecar eeeemdYe beuendery E339J39nditiD I39 yrr 0 m be eaeteiehed it l ee1 that 39 JT meuete equal here fer eeme peeitieve integer Hg 50 A 0R eel In eizher weiirde J1 must eque em ef the numfbeere p 4 9 3 hTheee veeluee ef A are cealled the ezigeen u39eIue393 ef the jprebeieme ElIde ELZIf E5pCII diI1g eelmticmne I1 I em 2 ein li3t E PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS 303 are called eigenfunctions It is clear that the eigenvalues are uniquely determined by the problem but that the eigenfunctions are not for any nonzero constant multiples of 4 say a sin x a2 sin 2x a3 sin 3x will serve just as well and are also eigenfunctions For future reference we notice two facts the eigenvalues form an increasing sequence of positive numbers that approaches 00 and the nth eigenfunction sin nx vanishes at the endpoints of the interval 0Jr and has exactly n 1 zeros inside this interval We now examine the classical problem of mathematical physics described in the preceding section that of the vibrating string Our purpose is to understand how eigenvalues and eigenfunctions arise Suppose that a exible string is pulled taut on the x axis and fastened at two points that for convenience we take to be x 0 and x Jr The string is then drawn aside into a certain curve y f x in the xyplane Fig 48 and released In order to obtain the equation of motion we make several simplifying assumptions the first of which is that the subsequent vibration is entirely transverse This means that each point of the string has constant xcoordinate so that its y coordinate depends only on x and the time t Accordingly the displacement of the string from its equilibrium position is given by some function y yxt and the time derivatives 8yat and 82y 812 represent the string s velocity and acceleration We consider the motion of a small piece which in its equilibrium position has length Ax If the linear mass density of the string is m mx so that the mass of the piece is m Ax then by Newton s second law of motion the transverse force F acting on it is given by 82y FmAx9t 2 5 Since the string is exible the tension T Tx at any point is directed along the tangent see Fig 48 and has Tsin 9 as its ycomponent We next assume that the motion of the string is due solely to the tension in it it FIGURE 48 304 UlF FFREHTIAL EEN1JteTIfHS As a teeneequtenee is the di ietenee EEEWEEEE the vatuee ef T sit at the ends inf eur pieetet Ft3ttI1El quot 39T sin EL ED p leeeemsee gt1 P 5 L 37 0 MT elm 9 m exg gi 6 the vibretietts are r eteteietve1y smelt ee that H is emelfl end Sign is eppreeaxeiemattely eqyuet te ten 3 then 6 yietlde awax m y st39 32 I end when tie e etwved te epprteeeh we ebtetin 5 e a j 33 b ym m ax SL1 R at Our pteeeet int ere esttt in this eqtuattien tie eent ned to the ease in belch G and are ee r1etetttI so tthet 0 equatiUt tt een be w ttert 3 fyt e T e 33 ea 3 Wtith H FUIE IT39 5EIlT15 that will emeetrge in the Pteblerrte equetiten SJ is eelled the eeets dt7mteee enmT weue eqeetien We seek a eeluetien ym 3EiIiS 39E55 the beundefry eemrlitiene emf D 91 and yimjrtil 0 H0 and the ineittiel eendJittietns 3 e e quotT II 11 e 5395 5 tr J and y x il ftx t 12 C39enditiemst 9 emtdf NJ express thee eestuemspttien that thee ens M the sttreitng are petmeneltentt39ly xed at the peitnte xi Flt and 1 1 and 11 and 12 assert that the strittg is mtettientese when is ereteeeted end thate y f e ie ehepet at that merrtentte We t1ete extptieittyt hDwequoteer that tneune ef these eenditiettts are in any way connected with the d etquotivetiet1 ef TA and E Wet shell tgiywe e fetzrmel te etluttien of B by the mesthed ef 3 ED rtIZt i nt ef earEetf e9t This emeunts te teeking f3t1quot setutiens ef the termt Jw E MittII 3913 wheieh ette tfaettetaible nite g ereduet ef rfunetiens eeehe eff which depends em enlyr enue ef the irtdependent war39tathles t Whem 113 is teut1etjiet utedt inter PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS 8 we get a2u xvt uxv t or r 1vquotltrgt ux a2 vt 39 Since the left side is a function only of x and the right side is a function only of t equation 14 can hold only if both sides are constant If we denote this constant by 21 then 14 splits into two ordinary differential equations for ux and vt 14 uquot Au O 15 and vquot Aazv 0 16 It is possible to satisfy 9 and 10 by solving 15 with the boundary conditions uO uJr 0 We have already seen that this problem has a nontrivial solution if and only if A n2 for some positive integer n and that corresponding solutions the eigenfunctions are ux sin nx Similarly for these z1 s the eigenvalues the general solution of 16 is vt c sin nat C2 cos nat and if we impose the requirement that v 0 0 so that 11 is satis ed then c O and we have solutions vt cos nat The corresponding products of the form 13 are therefore yx t sin nx cos nat Each of these functions for n 1 2 satis es equation 8 and conditions 9 10 and 11 and it is easily verified that the same is true for any nite sum of constant multiples of the y b sinxcos at b2 sin2x cos 2at b sin nx cos nat If we proceed formally that is ignoring all questions of convergence termbyterm differentiability and the like then any in nite series of the form yxt 2 b sin nx cos nat b sin x cos at nl b2sin2xcos2at b sinnxcos nat 17 is also a solution that satis es 9 10 and 11 This brings us to the nal condition 12 namely that for t 0 our solution 17 should yield PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS M These b are very familiar to us and are called the Fourier coef cients of f x With these coef cients 18 is the Fourier sine series of f x or the eigenfunction expansion of f x in terms of the eigenfunctions sin nx and 17 is called Bernoulli s solution of the wave equation The above solution of the wave equation is clearly riddled with doubtful procedures and unanswered questions so much so indeed that from a strictly rigorous point of view it cannot be regarded as having more than a suggestive value But even this much is well worth the effort for some of the questions that arise especially those about the meaning and validity of 18 are exceedingly fruitful For instance if the b are computed by means of 21 and used to form the series on the right of 18 under what circumstances will this series converge And if it converges at a point x does it necessarily converge to f x We give the following brief statement of one answer to these questions that is fully covered by the theorem proved in Appendix A at the end of the preceding chapter The function f x under consideration is de ned on the interval OJr and vanishes at the endpoints Suppose that f x is continuous on the entire interval and also that its derivative is continuous with the possible exception of a nite number of jump discontinuities where the derivative approaches nite but different limits from the left and from the right In geometric language the graph of such a function is a continuous curve with the property that the direction of the tangent changes continuously as it moves along the curve except possibly at a nite number of corners where its direction changes abruptly Under these hypotheses the expansion 18 is valid that is if the b are de ned by 21 then the series on the right converges at every point to the value of the function at that point The need for a carefully constructed theory can be seen from the fact that if f x is merely assumed to be continuous and nothing is said about its derivative then it is known to be possible for the series on the right of 18 to diverge at some points Another line of investigation considers the possiblity of eigenfunc tion expansions like 18 for other boundary value problems If we put aside the issue of the validity of such expansions then the main problem becomes that ofshowing in other cases that we have an adequate supply of suitable building materials ie a sequence of eigenvalues with corresponding eigenfunctions that satisfy some condition similar to 20 Suppose for instance we consider the vibrating string studied above with one signi cant difference the string is nonhomogeneous in 2It has been known since 1966 that there even exists a continuous function whose Fourier series diverges at every rational point in 0Jt 303 D1H EeermeL E 39UHquotFt ltHE the sense that ite Cteneiety m mfx may very frem peint tn paint In this situation 8 ie reelected by 321 emtx 3 22 If we again seek a eelutien et the term 5 then 22 tneeemes T 1 mAxtutx Ttrt and as hefetret we are led ten the fellewing beundary value Apreblem Lt JmJu D uat B What are the etigenvetues and eige n metien5 in this ease Needless te say we eemnettt gitvte pt39ECi answers witheut 1ntewing 5emethting de nitte abeutt the density funtetiern mx Hut at least we eent prewe that these eigtenttel uee and eigentunctiens texirst The detetts ef this tergument are giveen Appendix A at the end set this eha pte139t P 1 Find thee eigeIweluee An em eigetnfun etiren5 y fer the etquetiten yquot39 Ely 0 ieeeeh ef the fetewing eases stat em Um yt rfit 0 te em 0 ytzrrn 0 REE39 t39 IV VI 0 y1 0 t 310 0 3939L 1 when L 3 Q t A y L 0 L 039 when L L 390 F y er El D U wt1ene 3939 P Sehre the fettewing twe eretuteme fer39maIy rie wither eensiderirtg sueh j7fLti39El39 methtemeutieel ie5ue5 as the ii39fferent39tehiiit39y39 ef fttnetiens and the emwereeee erf series at ft Fun er FIGURE 49 PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS 309 2 If y F x is an arbitrary function then y F x at represents a wave of xed shape that moves to the left along the x axis with velocity a Fig 49 Similarly if y Gx is another arbitrary function then y Gx at is a wave moving to the right and the most general onedimensional wave with velocity a is yxt Fx at Gx at a Show that satis es the wave equation 8 b It is easy to see that the constant a in equation 8 has the dimensions of velocity Also it is intuitively clear that if a stretched string is disturbed then waves will move in both directions away from the source of the disturbance These considerations suggest introducing the new variables a x at and 8 x at Show that with these independent variables equation 8 becomes 82 y 0 8a8B and from this derive by integration Formula is called d39Alembert s solution of the wave equation It was also obtained by Euler independ ently of d Alembert but slightly later 3 Consider an in nite string stretched taut on the xaxis from 00 to 00 Let the string be drawn aside into a curve y f x and released and assume that its subsequent motion is described by the wave equation 8 a Use to show that the string s displacement is given by d Alembert s formula yltx Lfltx an fx am ltgt Hint Remember the initial conditions 11 and 12 b Assume further that the string remains motionless at the points x O and Jr such points are called nodes so that y0r yrt O and use to show that f x is an odd function that is periodic with period 2 that isfx fx and fx 2n fx C Show that since f x is odd and periodic with period 21 it necessarily vanishes at 0 and IE d Show that Bernoulli s solution 17 can be written in the form of Hint 2sin nx cos nut sin nx at sin nx at 4 Consider a uniform exible chain of constant mass density mo hanging freely from one end If a coordinate system is established as in Fig 50 then the lateral vibrations of the chain when it is disturbed are governed by equation 7 In this case the tension T at any point is the weight of the chain below that point and is therefore given by T mnxg where g is the acceleration due to gravity When mg is canceled 7 becomes 8 82y F9 gxc 9xl5t239 310 snIFsEsEHTsI E LLsTDNS Jr 391 FIGURE 50 3 rssusmE that this passtisal di srentiai vsquavtiun has a srsl39uIim1 sf the forms yxs E usixsr39I and ShE1W sis air mnssqusnss that um sssis sss ths nlilsuwing susdinssy di sssnstislli s1suatsissn ad v d 3 Au ng 39 If the irsdspsnd ent sarsissbEs is czllmngstd fmm 1 tn 5 p show that sqnsati sns quot s hssmmsss din sis L G 3 as U sw 1i h spars fmm anrts39tsiusn is Bssss s aquariums 129 far the spssisl sass in which p E P Smlss ths sibr39stisng sstrisnsg prnblsm in the tsstt if the inzitial shape 12 is given quot by the f unm tim1 fa fix Ess39M El 5 s 5 r 2 sss irjfss ssf 5 st 5 ss lib I 5 39r r I 1 PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS 311 x O S x S Jr4 c J174 Jr4 S x S 3Jr4 09 J17 x 3Jr4 lt x E Jr In each case sketch the initial shape of the string 6 Solve the vibrating string problem in the text if the initial shape 12 is that of a single arch of a sine curve fx csin 2 Show that the moving string always has the same general shape Do the same for functions of the form fx csin nx Show in particular that there are n 1 points between x 0 and x Jr at which the string remains motionless these points are called nodes and these solutions are called standing waves Draw sketches to illustrate the movement of the standing waves 7 The problem of the struck string is that of solving equation 8 with the boundary conditions 9 and 10 and the initial conditions 8 X gx and yx0 0 at 0 These initial conditions mean that the string is initially in the equilibrium position and has an initial velocity gx at the point x as a result of being struck By separating variables and proceeding formally obtain the solution 36 yxt Z c sin nx sin nat l where 3 c rrna I gx sin nx dx 0 41 THE HEAT EQUATION When we study the ow of heat in thermally conducting bodies we encounter an entirely diiferent type of problem leading to a partial differential equation In the interior of a body where heat is owing from one region to another the temperature generally varies from point to point at any one time and from time to time at any one point Thus the temperature w is a function of the space coordinates x y z and the time I say w wxyzt The precise form of this function naturally depends on the shape of the body the thermal characteristics of its material the initial distribution of temperature and the conditions maintained on the surface of the body The French physicist mathematician Fourier studied this problem in his classic treatise of 1822 Th orie Analytique de la Chaleur He used physical principles to show that the temperature function w must satisfy the heat equation aa2w 82w azw aw 8x2 8y2 822 T9 1 Y rnFFeaetmT39I39aL E 39UATi t 5 FiiGUHLE SI shalt rettraee hie teaeening in a simple enedinnenaietnalt siiteat1ieniaintd ttheretihy detirve the ene ditnenetiteinai heat eqgetatien ii39eiiiewing are the physiealt ptineieiee that will he n eerJee w Heat ewa in the diireetien ef decreasing temperature that is ffrvem het regiene tea eeici 1tegi enat h The rate at whieht heat heats aeteea an area ie prepeartitetntai te the area and the the rate ef ehange ef tetnteeeatete with I E SpEtE39I te ciietanee in a 39itLi1itTEttiti n perpetndieintiat te the area This preert 39tientaltity faeter is ditentetted by k and called the thetrm ei eeeeieetttrity ef the eubetaneeg C The quantity ef heat gatined er ieat thy a heed when its ttenttperatugtte ehtang5ea that ah the ehange in he thermal energyt prepettienal he the quotmash et P heady anei tie the ehange ef temperiatntei tuiThie prepetttie naiiithy flaeter is dented by r and called the epevei e heat ef the eehatanee We new eetneitcie ri the ew ef heat in a F eyiIintdrieaI fried et et eeeeeettienaii area V Z 51 wheee iaterltai enrfaee tie Apetfeetiy inetlitateid see that rte heat eets threeugh T Tlhje the ef the WtJ39fd GW means that the tentperaitutre is aeeernree te e nnit eirrn en any eteea heete tient and is therefere a fnnetien eniy ef the time and the pesitien ef the eteee eeetien say as witt We exaimtine the rate ef ehange ef the heat eentained in a thin eiiee ef the are between the peeitiene x and I e if is the deneiity ef the red that is its mass peat unit veiitutmett then tease ef the slice is V pA ex Furthetrntere if ewe is the temperatterie change at the peint 2 in a etnail titne intervatit mg then he teiie he that 0Z qeanytity ef heat ete t ed in the ehee in h tirne intaeanrai ie eff t em e we cert ext ewe ee the rate at which heat is being entered is appr39ehimiateiyi eH aw k t CD ht PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS 313 We assume that no heat is generated inside the slice for instance by chemical or electrical processes so that the slice gains heat only by means of the flow of heat through its faces By b the rate at which heat ows into the slice through the left face is kA 8x I The negative sign here is chosen in accordance with a so that this quantity will be positive if 8w 8x is negative Similarly the rate at which heat flows into the slice through the right face is kA 191 8x 1 xAx so the total rate at which heat flows into the slice is 8w 8w kA kA Bx xAx Bx x 3 If we equate the expressions 2 and 3 the result is 8 kA l kA595 cpAAxA V x HA 8x At or k 8w8xxA 8w8x AIf cp Ax At 39 Finally by letting Ax and At gt 0 we obtain the desired equation 82 8 2 39 lt4 quot939 5 where a2 kcp This is the physical reasoning that leads to the onedimensional heat equation The threedimensional equation 1 can be derived in essentially the same way We now solve the onedimensional heat equation 4 subject to the following set of conditions the rod is J17 units long and lies along the x axis between x O and x Jr the initial temperature is a prescribed function fx so that Wx0 fx 5 and the ends of the rod have the constant temperature zero for all values oft gt 0 wOt 0 and wJrr O 6 K314 I2JIFFERE h7TlAL EuLmTmHs W try frr a svlVutiam Eat this tnm1ndary waaillua gpmblem the m amptVhD d of sE paraVtim1 mf variabIIE that wnnilEd ac well in the cam cut the wave Eriquat tng that is we seek E S lutian Elif having the farm wxI ux1r W T W fhen this esxpressigtan is 5ubsrtituted in 4 the result can be wriumm 1 um ux 3 MI Since amh side mf this equati An d epeLn s an only me uf th variaMe5 bath sildeg mrust be cmn5tan1tj and if wa de ate ths mmmmn constant value by L than 8 Eljlit i TIEI the twnja mdinaI39y diHere11tial equatims Hquot D 3 and u la239u U V J ust as in Sectthinn 0 WE srjlve 9 I 1d Sa4ti4sfy the bunda ry CCIndiTEi MS 6 setting A 2 1 fm any 4prVsiquottiive inJtegar n and the c mr139E5pnVnding EigE f fllnE1fi is ux E 5i1mx With this value Erf eq uati nI1 ID bec mes 4 P L which has the easy snlmicrn u 9 quot2 Equot The FE SlllIIquotiLI391g gpmducits of 1116 farm are 39t heref nrE p Z n c II II mm This b rimg5 US to the paint where we lm w that each nf funct1inn5 ll saxtis ses equat39iDn 4 and the bmundiary EC1F di1iCi39 S 5 and it is clear that the zsame is true for any nite 39IinE if cm 1inatinn of the wj b1e quot 5iVn 3 bigquot quot I s3I1n2x F Vb ce39quot g g si11 fM a 12 Without dweiziling mm innp Urtan4tV mathematical issues of CUI1 39EfgEA EE and teImJbyaterm di EerrentiabVility we now pasts fmrn 12 ten 1113 rr s39p nding in nite semie3 K173 nLl will a smlutimm of our miginal bmundary wa1u4e jpr 1bAiem O it 3110 us m mtisfy Em ir1itial unndit39mn EL that is if 3913 il Ed UE39E5 EU the initiyai PARTIAL DIFFERENTIAL soumoms AND BOUNDARY VALUE PROBLEMS 315 temperature distribution f x when t 0 fx 2 b sin mc 14 nl To nish this part of our work and make the solution 13 completely explicit all that remains is to determine the b as the Fourier coefficients in the expansion 14 of f x in a Fourier sine series b J0nfx sin nx dx 15 Example 1 Suppose that the thin rod discussed above is rst immersed in boiling water so that its temperature is 100 C throughout and then removed from the water at time t 0 with its ends immediately put in ice so that these ends are kept at temperature 0 C Find the temperature w wxt under these circumstances Solution This is the special case of the above discussion in which the initial temperature distribution is given by the constant function fx 100 0 lt x lt Jr We must therefore find the sine series of this function which we can either calculate from scratch by using 15 or obtain in some other way see Problem 354 x smx f 400 sin3xsin5x Jr 3 5 By referring to formula 13 we now see that the desired temperature function is 400 1 1 wxt e 392 smx 36quot 39239 sin 3x 36 5 239s1n 5x Jr Example 2 Find the steadystate temperature of the thin rod discussed above if the xed temperatures at the ends x 0 and x Jr are w and wz respectively Solution Steadystate means that awat 0 so the heat equation 4 reduces to 82w8x2 0 or dzwdxz 0 The general solution is therefore w cx C2 and by using the boundary conditions we easily determine these constants of integration and obtain the desired solution 1 W W x quot39 WX 717 The steadystate version of the threedimensional heat equation 1 E39IFFEEEI393939TA1LEiZI U tT39It3H is aw SEWquot a gtw V p p as t 6 it is eelted LepIeee s reqeettatert The st udgr ef this equetien end its selutiens end uses theret ere matey eplieettietntst in the ttfheety ef grerwttettietnaeis a rich barenteh ef mtetttetmeties ea ed pettet1tieE Iheeary This tepie is C t 1ti1I1livEt1T in Appentdtis A at the end est thee nettt ehepvtert The teetrrestpezntding eqeatiaere in two dlimemsiteles is M U 17 as y this is es ssilutebt e steel if plane pzrebttemts ere under eensmetettietna Esqttatiett 1 else has a specie signi tcenee ef ixts ewe pP eemtplezt enet7Ejysitst N Demre the tihl39EE t ditWl39E L5i W3I tneet equetteieztt At the vedept i39rtg the restsKenitrtg it three test M the eese ef ts smell bee wiittt edges ex ey Aeeer1teineL ire e tegien itt ey39e aspeee wheres the tetmtpeteture ftmetietn w t39yeet is seugtjttt h C ildi ir the New ef Lheet tthteugh We l pp il feces ett the bees ZJt2t pIquotquott1diIJI3H I ten the l TIEiE then WEE quoteeist and ttte fy the eexist Q SeEts39e the beend etjt eelue prehvllerm in the test if the relnditieints ere e teted freen SJ and 6 tee wtteD fist Pz1 wlTI ten wttt W2 Hints 39Wr itt e ert1I W3rt gtj end remtrtemsbezr iEsemtpEte Sttppa E that the fletetel sutfsee es the thi It read iiFI the tzeet is net mst1tetecJ but itnsteed tadzisttes hest intet the sutreundiegest If N etetenquots Lew ef eeelittg eppiies shew that the en e dimetesienstt lttest eqjruetien tlzweeerrtes where e is s positive een tstsntt end we is the tempere39ture ext thee 5uI39IiLi39u1 d7iI1gEt In the preeedinge prebt1em md wustI if tlrte ends etquot tthe red are lvtept at DquotC Me E end the ir1it ieI tereperettlre d4istrih239utien is fr sj 0 In Eseettplte 1 stutppttse the ends ef the tree are itnsttmted insteed being kept at 393939 Ct What are the new tlmutrt iety endi tiems Fiend the teITItpetett1tte wett in 5 this ease Iwy using E nIjt etemmen sense R Splse the Ift bt m ef tntdimg wtett fer the rd erith insuletedt ends et ex U Ejittidt A 1 see the fptee ediang hrehemt if the irtttielt tentperett1re sdiistribeltetnt is 39 gjl39V IEl7l by wJrttD J PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS 317 7 The twodimensional heat equation is pF U 52 ax 8y2 at Use the method of separation of variables to nd a steadystate solution of this equation in the in nite strip of the xyplane bounded by the lines x O x IL and y 0 if the following conditions are satis ed w0y 0 wJry 0 wxO fx lin wxy 0 42 THE DIRICHLET PROBLEM FOR A CIRCLE POISSON S INTEGRAL We continue our overall program in this chapter of acquainting the student with important mathematical problems related to both partial differential equations and Fourier series Even though we cannot treat these problems in the depth they deserve within the limitations of the present book at least it is possible to convey an impression of what these problems are and brie y describe some of the standard methods for dealing with them We begin with the twodimensional Laplace equation mentioned at the end of Section 41 In rectangular coordinates x y it is 82w 82w O 1 axl ayl and in polar coordinates r 9 it is 692w 18w 1 82w 37a 33a3 39 2 It is an exercise in the use of the chain rule for partial derivatives to transform these equations into one another see Problem 1 below Many types of physical problems require solutions of Laplace s equation and there exists a wide variety of solutions containing many different kinds of functions However just as in the preceding sections a speci c physical problem usually asks for a solution that is de ned in a certain region and satis es a given condition on the boundary of that region There is a famous problem in analysis called the Dirichlet problem one version of which can be stated as follows Given a region R in the plane bounded by a simple closed curve C and given a function f P de ned and continuous for points P on C it is required to nd a function wP continuous in R and on C such that wP satis es Laplace s equation in R and equals f P on the boundary C PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS 319 n 0 the only suitable solution is v a constant and for n 1 2 3 the solutions of 4 are linear combinations of cos m9 and sin n6 v9 a cos n6 1 sin n6 We next set A n2 in equation 5 which then becomes 2 r2CflrL2 r nzu 0 This is Euler s equidimensional equation Problem 175 with solutions urABlogr ifnO urArquotBr quot ifn123 where A and B are constants We want ur to be continuous at r 0 so we take B 0 in all cases and we therefore have ur rquot If we now write down all the solutions w 2 urv6 in sequential order the result is as follows n 0 w a constant ao n 1 w ra1 cos 9 b sin 6 n 2 w r2a2 cos29 b2 sin 29 n 3 w r3a3 cos 36 b3 sin 36 It is easy to see that any finite sum of solutions of Laplace s equation is also a solution and the same is true for an in nite series of solutions if the series has suitable convergence properties This leads us to the solution 1 X w wr9 00 E rquota cos n6 b sin n6 6 n1 If we put r 1 in 6 and remember that we want to satisfy the boundary condition w16 f9 then we obtain 1 f6 2a0 Z a cosn6 b sin 19 7 nl It is now clear what must be done to solve the Dirichlet problem for the unit circle start with the given boundary function f6 and nd its Fourier series 7 then form the solution 6 by merely inserting the factor rquot in front of the expression in parentheses in 7 Of course the constant term in 6 is written as 11510 for the sake of agreement with the standard notation for Fourier series p lItFt FEiHE NT l eeust39ews EsempIe Seve the Diriehlet preli tem fer the lquotti i E eirete if fm 1 en the step hstf Cut the eirehe 1 4 H I5 and fHquotJ t 1 test the hI CEE ll39i391 hell etf the eirele rr J 5quot 2 J with frUquot fti rr 2 If Safe39t iquoten We hnew fmrrt fPteht em 354 that the Feuriser seri ets sfer fie is sih3H sin 5H T 5 A The selutiertt ef the DirieZhl3e t pareebteem is tI39lET E flEJ3939E 4 1 1 W rH39 quotr sin 8 Erquot 3H grssin 55 A PE b pB The diseessien given ehese is e39U EEI ed mesttly with fermsi pteeedurest and net with deliieetge qfuestietns erf EiDnVEf gEI39M2E Hwever we state witheet meetquot that the pz d b are the Feeuttiere eeef eients ef M6 then the series 6 eetnverges fer D S r and its sum wr 9 is e setutietn eat Lspleee s eqttatien in this regime Fefr this to be tme it is net necessary te sssume that fEt is eehethtueus er even that its Fourier series Ieenwerges It is Et IDiltJg 39l1 te assume ethst flf is inetegrshIe Ferthermete even sweith this week hypthests it etsums eut that fh6l is ttt llrtdi lrjf vsltte ef w rHj in the sense that we 9 N at every PCl39i1391 t39 ef CU39 tinui39ty of the funetien These remarkable facts have emergedfi hem earetelhli theere tiesi sm dI391es ef the PquotEtiSSD integral whizzh we new brie y desetile 3quot The Peisseta int egrsI The Ditiehlet pmhletm te r the unit C i1l IChE is new sewlved at litEt i fehrmsMy Hequotweuer s simplear erepressien fete this selutien een he tfeund as feilews if we elevrft mind a hit est teslesutftasttng with eemlex numhers As we i 1 tCJ39W the eeeh eients in 4 are giten by the fertrsutss e Jfet tetetstmfv dew bi tfgb sin Whern these are suhstiteute d in E1 then by using the i dEIquottI iIF eesm 0 we 9 ms w sin sin EMZUTE detsiIs en these i ntverest39mg me1ters er tIte ury sen be tteuhel in H 0 Csrstsw 39rtrr39eehret t39ees re the Theery ef Ferm39er s Series ens fntegrrefs 3eLiS MeeIttiiIart Lendett Ji9393UJ pp E5hU2574 Seelefy AH J et39r edm39sEeltn re Feer39i sr Eeries ens fntegr eis X n Bettjttt39nin New r e39rk 0 pp 39t e19 er pp 39413vE M2 ef the heath Bf Ss Negiy mentienee iin See39titmz 3 d PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS 321 and interchanging the order of integration and summation we obtain wr6 Jlt tfdgt rquot cos nB v dd 8 To sum the series in brackets we put or 6 q and let 2 re rcos at isin cr Then 23939 rquotequot rquotcosna isin me and 2 rquot cos na real part Z 2quot 1 I tgtlr real artr l 1 P 2 1z 391z 212 1z1 Z 21 z2 1z2 1r2 21 zl2212rcosar239 real part real part By substituting this in 8 we obtain wr6 i 1 1r2 f d 9 Zn 1 2rcos6 r2 39 This remarkable formula for the solution of the Dirichlet problem is called the Poisson integral it expresses the value of the harmonic function wr6 at all points inside the circle in terms of its values on the circumference of the circle It should also be observed that for r 0 formula 9 yields wlt06gt fr Ch f dlt1gt This shows that the value of the harmonic function w at the center of the circle is the average of its values on the circumference NOTE ON POISSON Simeon Denis Poisson 17811840 a very eminent French mathematician and physicist succeeded Fourier in 1806 as full professor at the Ecole Polytechnique In physics Poisson s equation describes the variation of potential inside continuous distributions of mass or electric charge just as Laplace s equation does in empty space He also made important theoretical contributions to the study of elasticity magnetism heat and capillary action In pure mathematics the Poisson summation formula is a major tool in analytic number theory and the Poisson integral pointed the way to many important developments in Fourier analysis In addition he worked extensively in probabil ity It was he who named the law of large numbers and the Poisson H IFF39EREHTIiuL Enuxmnzws dii5tributim1390r law of small nurnbersahas many appiiazatrinyns In Such phaannm V BFI as tl1e i5trifhiutiQ n at bland cf Els U1 3 micrns139UquotpE slid i of aut nm bil g an a high39wayi of cuStmer5 at a t11 eaterr ticlml t etc Aca rdisng In h l P iS5DI39I39l wag ra sIm r3939jL lump man His family tVIi el I enmumraye in many dirEElti IDlT1S from Ming 3 dDctm39 In being a IaWjrEr IIfni5 last an the th l fy that ramprh p5 ht wag for nJmhing heiIer but at Iias39t ha found his ninhE as a 5cienti5t and preauced wer wmks in a rtlativeljr shm t l1iTfEti m1e quotquotLa ri 2 sat la tranrai1 iLri39Iquote is wLrrrk39 he 5aid and he had gwd reasan mu Irrmrw 1 nquot 39v K Shaw that than DirihEampt pmblem f r139 Elie ir e xi 32 If W 39 Fy Gr 7 Awit h x r 05 E and rsin E slmw that aim 3173 0 333 2 g WW A r 5 3r 83 r F Srquot 33939 63239 3E39H STE W P r sin 8 ii r ms 6 3 y T p P 3 SW T San1ntaIly cuzmpute Er 0 and SDWE 2rhe Dir39iCh1E 3139DbE Em fmquot the runit circle if the biDunfJar 39 fUl39l TEiU39139fI fI i5 da ned by jagsf3E39 2 Has P 3 E 5 IE re1 in ii 6 r 39 IE 0 far 3 5 E c U f ru r nr 1 65 0 2 11 E 9 n f sin I9 for 0 E E 5 IE PD 1 fnr ii E 51 1 fi W3 53 M il 2 R5 where 1 H is the quoth nundar funmztmn has the sr uIin 1 quot 5 B wrB Ea an nnnH bquot SinrB i T where E and by are tilt fFm1IiEr nquote iciants crf f P alga that the Puiss1an inmgra fur 1hiI5 mnreV EmErail aasnE is 1 quot H 1 2Rmas1w r1quotrW M 39Let MP harm nic in a nane I39engimAn and Net C be any rirule I 339lquotmI39i1quotBlF cn1 mtainm1 in reirn Ftrmee that me mlue nf W at the emer If C is HIE a rtrage uf its values an ithe cia39cumfErm1ce TEES a majwur tlmuzxrem nif pn39tentia thezonr dus In Gnau55 PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS 43 STURMLIOUVILLE PROBLEMS We return brie y to the discussion of eigenvalues and eigenfunctions at the beginning of Section 40 Our purpose here is to place these ideas in a broader context that will help make an easier transition to the topics of the next chapter As we know a sequence of functions yx with the property that fa ymxynx dx O ifm 93939 n 1 an 95 0 if m n is said to be orthogonal on the interval ab If a 1 for all n the functions are said to be normalized and we speak of an orthonormal sequence A more general type of orthogonality is de ned by the property 0 if m n a at 0 if m n b qxymxynxdx 2 In this case the sequence is said to be orthogonal with respect to the weight function qx Orthogonality properties of this kind are possessed by the eigenfunctions associated with a wide variety of boundary value problems Consider a differential equation of the form d d 5 oltxgtf uqltxgt rxy 0 3 for which we are interested in solutions valid on the interval ab We know from Theorem A in Section 14 that if px p x qx and rx are continuous on this interval and if px does not vanish there then there is one and only one solution yx for the initial value problem in which we arbitrarily assign prescribed values to both ya and y39a Suppose however that we wish to assign prescribed values to both ya and yb that is to yx at two different points rather than to yx and y x at the same point We examine the circumstances under which this boundary value problem has a nontrivial solution Example 1 At the beginning of Section 40 we considered the special case of 3 in which px qx 1 and rx 0 so that the equation is y y 0 The interval was taken to be 03 and the boundary conditions were y0 0 and yrr 0 We found that for this problem to be solvable A must have one of the values ln2 n123 324 mFEEREm 1L A uumilows and that curr pnnciing sMuVtiDn5 Elir mg sin WZ W call d ma EL Eh EEgen1m ue5 Inf tha prnbfiem and the ynfx 35 cDrrE5pmJndimgJ e gEr1fun rfans In Vtha case at 111 mare gmewral muxatixznn 3 it mrns nut that if the functinms x amd qr are restricted in a reasotnahle Vwa3x spe ci VcaIly if P 3 I3 and qt U 0n ab thcin WE wi1l 3154 be able in Dtft in nDn trivial mlutitrna satisfying suitable bmlndary cDndi tinns at Ih twa c39EisIimt paints a and b if and mn jy if the paramet r takes cm certain speci c quotvaluas Thase are the e g3n1m ue3 Lf the boundary value pmb1an1 they are real numbers that can ha ar1rang d in an ixncreamng SEq39 E FlGE s 1m aI3v2 iI3 a39quotL igL quotAn4t39qquot39J w and furthermmre 1 a D0 as H This Dr fi g is desiArquotabEe b u it Bnales 115 t arrange the mr reapm ding E gE furIc39 n3 y1I MI Y JMI A 5 in their awn naftural Dl d 6139I A5 in ths case of Example 1 the eigenfunc titans am mm uniqfu but with th b undaryr mnditi ns we will be inmeri3st ed irI the3 aare deE3rminrampdJ up 10 3 I1D1nZE139WEl39l mnstant factm We new llmzmk fem pVcssible rthDg 3na1it3r jpmperries 0f the sequmrze sof EigenfIunctiCn5 G and in the pr cess Hf doing thi we will discmE139 what typ s of boundary conditimms are 5uita4bEeVquot 39IZZ0nsi1ier Hm vd1i FEf39EI11EiB31 Equ tiiun G writtan down for tw di amprEnt Eigerwalues Am and Am with jam and ya the wrr39espunding eigenfunctin3 if q Eta dx AM rAFm U and d a39yn P 21 9 F dip it O Mar FM D If we Eh39iff m th mmie cnmpact prime nntmi n far d3Iivatquotixre5 than an multipl3ri4ng the r51 Equali n by y and the seemnd M JP m and subtMriavtingj we find that 39 J yn i M21 yAmy I N 1rquotWm ya D We Imw move the rsit two tE lquotlquot 5 I0 the righgt and mftcgrate fmm 2 to b PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS P using integration by parts to obtain b Am qymyn dx b b I ympy dx I ypyZn dx lympyi i by npyZ dx lypy nlfZ Jby2py39n dx pbymbyZb yby nbl paymay2a ynay al 6 If we denote by Wx the Wronskian determinant of the solutions yx and yx which is de ned by ymx y39nx yx y x then 6 can be written in the convenient form Wltxgt yltxyltxgt ynxyrnxa Am An bqymyn dx pltbgtWbgt pltagtwltagt 7 We point out particularly that the integrations by parts in the calculation 6 and the consequent cancellations are possible only because of the special form of the first term in the differential equation 3 We want the right side of 6 or 7 to vanish so that we can obtain the orthogonality property b I qyy dx O for m a n 8 By looking at the right side of 6 we see that this will certainly happen if the boundary conditions required of a nontrivial solution of 3 are ya 0 and yb 0 or y a 0 and y39b 0 Each of these is a special case of the more general boundary conditions ciya c2y a 0 and do5 d2y b 0 9 where C or C2 0 and d or dz 0 To see that these boundary Di ferential equations having this specialform are called self adjoint See the problems below for an explanation of this terminology A315 IIlFFERI1E39JTlI L Emumeeeewe selu zienfs ym jx eamed yn x berth eatisfy the ret CD ditii 39 9 50 that eeeenditiens rteeliy de illfl glDE the righi side ef W v eaeish euppeee that the quot339a n39 p 0 lIyzr S ez a U Sineee this eysteme has a menetrievijeel eelutien irh C3 the eeee fieieeent determinant emuet v1enishee svfmIa yHle ge Me Similarly we e D and it fewlleews fmm this th i the rigm sidee ef T vanishes Beundary eeendzitiene of the efeerm 9 are eelleed heeeme geereeeue beuedeerye c ndi39 ie39m e Their 5peelt iaE feature the feet tehe39t any sum of eelutien5r ef eequetien 3 that i1deiwifuelF eeeteei5fy such b ned ffquot 39CU djquot ltienes wii else satiety the same beuendeery C39Ui1E 39Eei 1I5 Any deiffeerenteial eque Ieie139m ef the term with heeernegeneeus beundeary eeindeittienee is eale1e d e Srurree e ef i39 U Wf 4pre bs39397eltrne The Siigni eanee ef these ideas is that the enhege nee it3r prepertyL 8 piveee MS a fermel methed fer miein g series expermsieens ef fume enes f x in terems ef the eigenefuneetieeene ef Sw h a Sturm eLeieemneviMe pmblem Fer malty we are led in the feMe wing preeeeldurei We eseurne that R can be written in the f fme mm r 9 0 u we Meuiltiplying beet39h sides ef 0 qrynx and lntegreting terhm by term from H be yieeid e We L Ff enfryeir fir k k E H I exey Iyaxee 611 I q reLMIJEedr quotA V e 2 ea F rjryn dJj D Abeeceeuse eef fSI With the eeeef eeiems en determeineed by 111 feremuie ele393I P3 uzellsed an Keigenfninmuln E sI E IFiFLEQLIPI If A very impUr39t J m39 matfheemetiAcaE qfueeetien new erieeee that is fmiliar quottee us from Chapeter 6 and the eeee ier seetiens ef this eehepeteraeheW de we knew that me series 10 withe ceeef eientes determeeinlede by VIM ree ly repre5eeetex And what edees represents rneen39 Bees it mean in the sense ef epeimwise eemFergem ee Dr mean eeneveeergeneee Or perhaps eamrlee emer eeeneept a1ltD39gEsfh ff We hm39ee seen in Chepteer 6 how deif eulte PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS 327 some of these theoretical problems are for ordinary Fourier series which are the simplest of all eigenfunction expansions Two further special cases that are particularly important for applications to physics are concerned with the orthogonal sequences of the Legendre polynomials and the Bessel functions These two sequences of functions and their properties and the associated eigenfunction expansions are the subject of the next chapter Selfadjoint boundary value problems of the kind described above are called regular because the interval ab is nite and the functions px and qx are positive and continuous on the entire interval Singular problems are those in which one of these functions vanishes or becomes infinite at an endpoint or the interval itself is in nite Unfortunately many of the more important problems are singular and the theory must be correspondingly more complicated to cope with them5 Example 2 Consider the important Legendre equation in its selfadjoint form 1 d Zx1x2 lyO ISXSI Here the function px 1 x2 vanishes at both endpoints No bound ary conditions of the usual kind are imposed at the endpoints x l1 but it is required that the solutions remain bounded near these points It turns out that this happens only when A nn 1 for n O 1 2 and the corresponding solutions are the Legendre polynomials Px The details of this singular selfadjoint boundary value problem are found in Chapter 8 Remark We have done little more in this section than acquaint the student with some of the issues in this subject and we have certainly not provided any substantive proofs One of the rst questions about any selfadjoint boundary value problem Sturm Liouville or otherwise is this Does there exists an adequate supply of eigenvalues and cor responding eigenfunctions For the reader who is interested in these theoretical matters a full and rigorous proof of this existence theorem is given in Appendix A but only for a somewhat special case of the regular Sturm Liouvi1le problem described above NOTE ON LIOUVILLE Joseph Liouville 18091882 was a highly respected professor at the College de France in Paris and the founder and editor of the Journal des Math matiques Pures et Appliqu es a famous periodical that played 5Full treatments can be found in E C Titchmarsh Eigenfunction Expansions 2 vols Oxford University Press 1946 and 1958 and in E A Coddington and N Levinson Theory of Ordinary Differential Equations McGrawHill New York 1955 S l q 39 V PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS 329 that are provably transcendental One of these is 3 1 1 3 1 10quot quot39 10 102 10 His methods here have also led to extensive further research in the twentieth century7 011000100 PROBLEMS 1 The differential equation Pxy Qxy Rxy 0 is called exact if it can be written in the form Pxy 39 Sxy39 O for some function Sx In this case the second equation can be integrated at once to give the rst order linear equation Pxy Sxy c which can then be solved by the method of Section 10 By equating coe icients and eliminating Sx show that a necessary and suf cient condition for exactness is P x Q x Rx O 2 Consider the Euler equidimensional equation that arose in Section 42 xzy xy nzy 0 where n is a positive integer Find the values of n for which this equation is exact and for these values find the general solution by the method suggested in Problem 1 3 If the equation in Problem 1 is not exact it can be made exact by multiplying by a suitable integrating factor 1x Thus ux must satisfy the condition that the equation uxPxyquot pxQxy axRxy 0 is expressible in the form uxPxy Sxy 0 for some function Sx Show that ux must be a solution of the adjoint equation Pxquot 2P39x Qxl lPquotx Q39x RXlu 0 In general but not always the adjoint equation is just as dif cult to solve as the original equation Find the adjoint equation in each of the following cases a Legendre s equation b Bessel s equation C Chebyshev s equation d Hermite s equation e Airy s equation f Laguerre s equation 4 Solve the equation 1 x2y Zxy pp 1gt 0 xzyquot xy X2 p2y 0 1 x2y xy pay 0 yquot Zxy Zpy 0 yquot xy 0 xyquot 1 xy py 0 3 yquot 2xy 4y0 x by nding a simple solution of the adjoint equation by inspection 7 An impression of the depth and complexity of this subject can be gained by looking into A O Gelfond Transcendental and Algebraic Numbers Dover New York 1960 t2IIFFEFtEHTtAL EtfjU39AquotI IID39N5 10 Shaw that the dj i1t39I1Z nf Thl adjt1ittnt f ttm liqu ti tPxtquot39 QrjI yquot Lxtjy U is th rigina Equati t1 Tltm ttquattimn t Q Jty39 P U is EaH etl tEEfaa39djt1E39nt if its EdjDi is the same equatinn IEt MtEEPM f ilf nntattun 3 Show that this tquation tsatttftaadjntint if and Dt tljquot if Pquotx Q xt In tE1lijSJ case the EqLi tquoti EJi bezmmm Fwyquot PtI39 P Rtttt tt 0 or HAPAEIDVET REX D39 which is th ttantdard ftmm of ta saif adj int eqtjatimrri b Whsich of the Eqtlattit nts in Frtrbltnt 3 are s elf atdtjntn t Shaw that an3r Eqgu ii t Pfxjm Qry R xjyt U can be m di sailf adjtImt by mutttiplying thtrmtgfht by t L E15E aquot391FJEil P 39USittg Pmhtarn whctnt nc e5sarjg u pttt each equation in Pmbtem 3 int the standard 5El39f lEjClit1tPI flttrm dtzstzrizibed tin tPrnblem Cttrt5idtBr t hEt tEgttIar 3t utirnfLi utti11E pmhtem t i fi lg at equatittn 3 with the bnundary 1DI1dii i39li139I1 E 9 Prime that E irquot39Et39 5t e39ig infun cttmn 0F 39utniquE ctcegpt far a mtt5 t n39t 39fattt3rr Hint Let y iI I 1 and Et 0c B eig ettf uttcttittts ut r39resp0ntd tng ta 3 singte eigctntsat1ue 0Z and use their Wmnskiant tn sihtzuw that t tt jpt are linearly dep ncnt rU1I1 t gb 39 I SidtET the fttIOwing saltadjttin39t htmtndatrr ttzaltte ppmhtem an m5 5pm Wtttf1t rtxm t Nit t Mb t 39 tJvquot tbt WhE1E ptt Mb It N assumed that pt p39t qx and rr4 are CquotDE1 I Ut H5 and that Apt x 2 U and q xft 2 U fttr a 3 I E b This prC139b1E39tTt tt1en said tn ihatrc patrt39m Et butttd5ary E mdt39ft 39 5 It can be pmvetd that t I39t 39tquotIE Ei ti i a Eit q39uBrIl3E f eiitguenva1uE5 and EL lt14L1 q 12 s at 0 E 3 54 such that lim Aw 0c 3 By attart1inrittg tlhye Cacwla t 39 tn r t3 Eh w that EigE f i1CEiDi S ED39IEfp f d ing quotED dtstttnctt etigentratltttes are n139tthngurnal with r spetat tum t39hE weight fUtlvUEiEtn Q In this tzas3 hmwevert to eatzilt igenvaue 39thtre may t t39t iEp39D nd EiI39 I39tEI39 one tilt tw3 Iiinctttmr i gtdEIpKEntdEE tt EigEt39tfut1CIiI 3It1S t vquotEritfur this by nding the aiganvaltrms and CDfI EtSpU 39dit tg E IEE39 fu miEm n fat the fDblsEmi yquotquot39 y 39039 39ttthet39t quot 3391 Jit and y 3939 t39i quotquotquott x can f3rIU39quotDi1EtII tmt hate tttma than IWD Lifrtdepettt39tvd aertt EigEr t f1tfI t t39 t39i Its a55i att1d wt th a tp tquotltCUtli1139 tig 39t1tt39att1tE39 lib Cy PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS 331 APPENDIX A THE EXISTENCE OF EIGENVALUES AND EIGENFUNCTIONS The general theory of eigenvalues eigenfunctions and eigenfunction expansions is one of the deepest and richest parts of modern mathe matics In this appendix we con ne our attention to a small but signi cant fragment of this broad subject Our primary purpose is to prove that any boundary value problem of the form 4023 which arose in connection with the nonhomogeneous vibrating string has eigenvalues and eigen functions with properties similar to those encountered in Section 40 Once this is accomplished we will nd that a simple change of variable allows us to extend this result to a considerably more general class of problems We begin with several easy consequences of the Sturm comparison theorem quot Lemma 1 Let yx and zx be nontrivial solutions of y qxy 0 and z rxz 0 where qx and rx are positive continuous functions such that qx gt rx Suppose that yx and zx both vanish at a point b0 and that zx has a nite or in nite number of successive zeros b b2 b to the right of be Then yx has at least as many zeros as zx on every closed interval bb and if the successive zeros of yx to the right of b0 are a 12 a then a lt b for every n Proof By the Sturm comparison theorem Theorem 25B yx has at least one zero in each of the open intervals b0b bb2 bb and both statements follow at once from this Lemma 2 Let qx be a positive continuous function that satis es the inequalities 0ltm2ltqxltM2 on a closed interval ab If yx is a nontrivial solution of y qxy 0 on this interval and if x and x2 are successive zeros of yx then It Jr Mltx2xltr 1 Furthermore if yx vanishes at a and b and at n 1 points in the open interval ab then mb alt n lt a Jr Jr 39 2 PARTIAL DlFFERENTlAL EQUATIONS AND BOUNDARY VALUE PROBLEMS 333 appeal to the inequalities 2 which in this case become ximb alt ltI Mb a 77 n It or nznz lt A lt n2Tl392 M2b a2 quot m2b a239 Equation 3 is the special case of the Sturm Liouville equation 1pxgtfjf Aqmy o 4 in which px 1 We assume here that px and qx are positive continuous functions on ab and also that px has a continuous derivative on this interval If we change the independent variable in 4 from x to a new variable w de ned by dz wx I 9 a pt so that L 1 and y 1 1f y dx px dx dw dx px dw then 4 takes the form d2 51 35 lq1wy 0 5 where qw is positive and continuous on the transformed interval 0 lt w S c wb On applying Lemma 3 to equation 5 we immedi ately obtain the following statement about 4 Theorem A Consider the boundary value problem d d pxgtj mix 0 ya you 0 6 where px and qx satisfy the conditions stated above Then there exists an increasing sequence of positive numbers tltA2ltAlt1ltA that approaches 00 and has the property that 6 has a nontrivial solution if and only if A equals one of the l The solution corresponding to A A is unique except for an arbitrary constant factor and has exactly n 1 zeros in the open interval ab One nal remark is in order As pointed out in Section 43 we usually refer to 6 as a regular Sturm Liouville problem because the iaiFEnEH39rrmL EEUJ5nTiquotCiNS inienrviaili is 39I391ii E and the functiims p 39x and qrc are pnsi iive and 39C3i ZIiiI liIiI1l1liJiJi5 on thaw E39 Iii39E iiniiewaii iSiriguiar pmbiems arige when th intervaii is in rnite Elli Whi nite anal pri mgr qr iJ ni5ihEE Gr is idiiS39Ci TTt iI1iCl iU5 at mvrie mt bath sindnainisi Tl1gtesi pimbiemsi are E Di7ilS39iidEl ab39I j39 more dif uIi and 0f m1r5ei are mm c t3ver eed 0 Dhiif diisciiusisiimn in this appemidixz Unfnriiuriatelm imamr tih most iiquotitEiI39BStil391g di fereinitiiai EqI1a39i n5 are siinkiuiiar in itiimis sense We I1 iEfI391IiIJI 1 Legmdre 3i quairimi E d 39ChHbymEu E vequmiirgiri 1 df 1 P x2 y i 5L U 15 351 dxi QM Nady T R V i I A 2i ni 1 1 dxiil x dM 1 M y am is quot3 EqM Ef Eequot3939 ii quotquotEr U m 7 JLT lt m and Laguerre 3 aquatiiim Ei5quot Viequotiy E 0 Ir p 1 lt5 p Tihiese eqiuaiinns appeairied in Cmpter S whmei they were studiei fmm an eniirely di ieirent point of view CHAPTER 8 SOME SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS 44 LEGENDRE POLYNOMIALS This section and the next are entirely devoted to the technical task of de ning the Legendre polynomials and establishing a number of their special properties It is natural to wonder about the purpose of this elaborate machinery and more generally why we care about Legendre polynomials at all The simplest answer is that the Legendre polynomials have many important applications to mathematical physics and these applications depend on this machinery For the bene t of readers who wish to see for themselves the physical background and several typical applications are discussed in Appendix A There is another answer however which is less utilitarian and applies equally to our subsequent treatment of Bessel functions It is that the study of speci c classical functions and their individual properties provides a healthy counterpoise to the abstract ideas that sometimes seem to dominate contemporary mathematics In addition we mention several items that arise naturally in the context of this chapter which we hope will be of interest to all students of mathematics the gamma function and the formula 335 nI1sFEREHfrLam EUUATILUNMS Lamberfs D 39Ii 39UEd frati m if r Itl1e Itangtintt I p w p w y and thhwe famcaus series 1 E n1 E ea l39l A and V E P whuse sums were discravexed by Euhlher in thheh asarljy eighteehnthh cehnthuryh and which app ar again in a surprishihg way in mnhecIih with the EEFGS of Bessal functhrEhn5h Naw fur tha Leendre pDlyn mi ls hthehgmmhres which we apprnach T way 0f the hypehrgheh mmrhic equati n 39 In Swtihhnh 0 we hushed Legehndrhe 5 E quatihnh ta illrushthrat3 the ishnique if dhih g pcnwer szeriehs sutzellutinns Tat uhrdihzarjy gpnihnhtsi AFW reimahnsh ehxp1aincd in Ahjppehnhdix A5 we aw wrhite Ighis equathi hn in the f rimh 1 1tf2 39 nrn s ID L whheere av is rund rstmd m be a l391D l397l17lJ Eg3 Eih 1rquot hinteghe1quot hTquothe reader wil re aM thhat ah the ssghluhtmnsh Bf M fnund in Seminn are analytihc on the quotinternal 1 7 1 1 h awrevher the S l1l ti n5 WDSEIZ useful in the applica 1riorls are thusE bounded hear 1 and fur chmnvcnhi cnce in singlin thheseh 0111 we change hthe ihdtphenhdEn1 variahhl e fgr om 3 two A hgh I mahkesh Jr 1 E fIA39TEiSp39 i11 j In I 2 U and trahnsforrms 1 into m1 mwu2mwmnnym gm wher 3 the pr39himhes signihfy derhivatives IEEPECI to E This is a hmperigeU mEs39thric equatin with H n b 2 Hf L and L 3 1 50 it has fUl IIW39iI391g pEa E3namihal shlutinh nhehahr I U yFhmnLLm M AdhIhhihh Marie hLcgen aIre lTr3952h 1333 nnumn Iermhd his prn1 nnmials in his rcsveahrch an the ErEViI 39EiLi l attra 39tian altquot ellipsiuhids He was 21 very good F1 tIfIEl 1 mathhrha39icjam whh had thzz miafhnrtunsn mi sensing meat of his Mst wm k ih lhpti intngrais humber Ehcorly arldi Sh n1 zhthhd of I235m scwaeres 5upEr5Ed cl by the whimrmc nt5h DI jmunger mad cams mn Fm instance ha deviated y39cars in his rlte5earch an ltipti iln1I LEgr3 i5 and his awnmiume ErnaaEisE canu I11 suhjvact had 539ahurr22 app EaIEIl in quotprim when lhh 2 discmwri5 umf Aha and Jacszrhh1 r39E 39 DNhlLIEin IliiEI39E Arm eld It1mph39tc39 HE was vwgry r39Em arhahh fwsar the ggheru39u5 5plth39iit with which 1hewrcpeatc dE3 wglhmcd summer and lm39ter wmh thhat made awn uiibsulhcteh SOME SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS P F Since the exponents of 2 at the origin are both zero m 0 and 7712 1 c 0 we seek a second solution by the method of Section 16 This second solution is y2 vyl where 61104 1eI2 1rl dt yi 1 1 1 2t1 2 J 1 I t 11 t by an elementary integration Since y is a polynomial with constant term 1 the bracketed expression on the right is an analytic function of the form 1 a1t aztz and we have 1 U 39tquota1a2tquot39 This yields v logt at so yz ylogt at and the general solution of 2 near the origin is Y C1 C2Y2 4 Because of the presence of the term logt in yz it is clear that 4 is bounded near t 0 if and only if c2 0 If we replace I in 3 by 1 x it follows that the solutions of 1 bounded near x 1 are precisely constant multiples of the polynomial Fnn 11 1 x This brings us to the fundamental de nition The nth Legendre polynomial is denoted by Pix and de ned by nn 11 x 1 Px F nn 11 1 x 1 12 2 n n 1n 1n 2 1 x 2 202 2 n n 1 n n 1n 1n 2 Zn L W 1 x X 2 1 nn 1 nn 1n 1n 2 2 1 M x I 1 20222 x 1 Zn x 1 5 n22 U E iFFEE 1EH7rmm m1Ja T1nus We kn w fmm mmquot wmrk in Sectinn 28 that Fx is a plwmmial M dugm E n that mntaina n fy mram mr mn1y 5 fp wers Elf 1 awarding 1 is awn G1quot ind It can therefore ha writtem in Ihe farm Pnufx l auxquot a 31quot E r a 139 A where S sum ends with an 0 Art is even and air if 1 is mddw It is dear mm 15 Iihat PH1 1 fm every H and in aw 0f 63 we alsu hawa P 1 C1 As it S t NdE fm mula 5 is a vet inE0nvenient ml ta was in studyiI1g Ex an we ank for 5mmEtLhing simgpiiler We muldi e xpan each terrn in 5 collect Iika ptw rs of 1 and arTr angE the r sult in the ifnI m 6 but this would be unnecessarily IabDri us we shall dam is notice from 5 that 1 2nfrE2 39 and cg lcsu ate qua ai4 p rEcu r5iw3l in terms if What is named here is f rmul 25349 with p replacd Aby n and 1 r 2 m k3mkan H 1 i Mk is 2 Car II k 2n pq k E 1 when k f H 2 0 this ytields r1r1 E 1 1 FmF 22r1 1 av r T 3 42n t quot39E Mn r1 331a 39 242n amp 12n 2 0e an 3 an 4 and so on an 6 b econ1e5 2 Hfn L nI 22 i r 22n 1 WV W EMT 31 p wmn 3 Mn f P E 9 39 245 n2k T 2k12n 1M2 3 En 2 VV ML nu P 39xL nJ1 J 1 TE some SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS 339 Since nn 1n 2k 17quotE5 and Zn Zk 1Zn 2k 3Zn 3Zn 1 Zn Zk 1Zn 2k Z2n Zk 3 Zn 3Zn ZZn 1Zn Zn Zk ZZn ZZn Zn 1 Zn n k Zn ZkZquotn k 1n 1n quot Zn 2kZquotn the coef cient of xquot 2quot in 7 is n Zn ZkZquotn 1k n2Zn Zk Zkk n Zk Zn n k kZnn k n 2k 139 This enables us to write 7 as quot2 Zn Zk P 1 k n 2k X 0 Zquotk n kn 21 8 where nZ is the usual symbol for the greatest integer SnZ We continue toward an even more concise form by observing that W 1 Zn 22 Pquot n 2k X 02quotkn k n 21 W21 quotwk Li 2n 2k kzg N 7 N dxquot 1 dquot W2 n E 1 Z 39ndx 39 k0k n k ltx2quotlt 1gt If we extend the range of this sum by letting k vary from 0 to n which changes nothing since the new terms are of degree ltn and their nth derivatives are zero then we get 1 dn Zx2quotquot 1quot 2quotI l dxquot k Pnx and the binomial formula yields dll Z ndxquot x2 UN 9 H quotDIFFERENTIAL EfQLmTmM5 5wx I1 0 52 expVres5iwiism 39fUI 39 Pntt is cama Rm r39gues frampmu V aF It prmvides an AreIatively easy mmhmd far cm1mpu4liEng the mce3sive LvagEndre p ulynCr JT1 i3lE of whirzfh the rst few 52 are w 1 P x 4 EmJ mr 1m m wH 3m even easincr Jr CEd Lil39FE is suggessted in IPmibl em I2 and a mare signi nant ap piatian of 9 will appear in the next E EEtiU 0 L The funcM7uun on the ft Side ml 1 ii 39 x P I Dlinde RKJd IFigIaJE39S W9739d 1 5H was 3 Fr39rnM h 1arntr Whfl im quot tn the aid 039 i aut HEvI IIfi Saiinim Sir1nun thme I 7mt 1itlampr nf 5cuialJi m in his E39JEEIiI lJ lE malt P SI IjIiquotn3Ii lmili him d711ring W last years af his life and bmame Elna Hf Eaurpliesi discipl a He disrcmagreed aha ainrw r39m uIa in l ii w but 395E39aMlTI tnerE39aflcr hrEEHITIHE inmtmE5t Ed inrl Ih 539mnti m ga niquotzaiinn cjf 5nnrimL1r amzi ne39wraer riquotquotEurmZd rm znnathma1ic5 THE term Rndriigu35 fIr39rr1uIJE is GWEN applimJu by tran4ferniE E1 snmLar39 ExprE SSifi 5 far mher IEa5sisra p I 5 nI3urnia1iI5 DI W hi 1I39m RmdrigurE5 him aE knew namhing 341 SOME SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS is called the generating function of the Legendre polynomials Assume that this relation is true and use it a to verify that P1 and P1 1quot b to show that P2O 0 and P20 1quot139339 39 Q quot 1 2quotn Consider the generating relation in Problem 1 1 ED P quot V1 Zxt t2 go xy 3 By differentiating both sides with respect to 1 show that x 2 2 Pxtquot 1 2x 2 Z nPxtquot n0 nl b Equate the coefficients of tquot in a to obtain the recursion formula I1 1Pmx Zn 1xPx nPx c Assume that P0x 1 and Px x are known and use the recursion formula in b to calculate P2x Px P4x and Px Establish the generating relation of Problems 1 and 2 by the following steps a Use the binomial series to write 1 13 1 t2x tquot1Z t2xti it22xt2 13 k k nI rl l 2quotquot39n1 Zx 0 132n ln 2quotquot t2x t b It is clear that t can occur only in terms out to and including the last term written in a By expanding the various powers of 2x I show that the total coe icient of tquot is 13Zn1 132n 3n 1 2quotn r ax 2quotquotn 1 1 ex 132n 5n2n 3 n4 2quot 2n 2 2 Q c Show that the sum in b is Px as given by 8 This problem constitutes a direct veri cation that if Px is de ned by formula 9 then it satis es Legendre s equation 1 and has the property that P1 1 Consider the polynomials of degree n de ned by dquot 2 1 quot dxn x gt a If w x2 1quot then x2 1w 2nxw 0 By differentiating this equation k 1 times show that x2 1wlt3952gt 2k 1xwlt39gt k 1kwquotquot 2nxw 2k 1nwquotquot O and conclude that y w quot is a solution of equation 1 yx SOME SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS 343 and by continuing to integrate by parts we obtain 1n 1 I n 2 n 2quot If xx 1 dx If f x Px with m lt n then f quotx O and consequently I O which proves the first part of 2 To establish the second part we put f x Px Since Pquot x Zn2quotn it follows that I 1 x2quotdxa 1 x2quot dx 3 If we change the variable by writing x sin 6 and recall the formula proved by an integration by parts 1 coszquot 6 sin 9 2quot 2n 1 2n 1 Jcos2quot 9016 fcos2quot 6d6 4 then the de nite integral in 3 becomes 2quot It2 J2 I cos2quot 9d6 cos2quot 6d6 0 271 1 0 2n 2n 2 gr Qde 392n12n1 3 0 C03 2quotn 22quotn2 13Zn 12n 1 2n2n 1 We conclude that in this case I 2Zn 1 and the proof of 2 is complete Legendre series As we illustrate in Appendix A many problems of potential theory depend on the possibility of expanding a given function in a series of Legendre polynomials It is easy to see that this can always be done when the given function is itself a polynomial For example formulas 4410 tell us that 1 2 1 2 1 P0x X P1x X2 quot quotP2x quotP0x P2x 3 3 3 3 3 2 3 2 x3 3x 5P3x P1x 3P3x and it follows that any thirddegree polynomial px b0 blx SOME SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS p satis ed when fx is a polynomial but in the case of other types of functions we have no way of knowing this and our conclusion that the coef cients a in 5 are given by 6 is of doubtful validity Nevertheless these formal procedures are highly suggestive and can lead to legitimate mathematics if we ask the following question If the a are de ned by formula 6 and then used to form the series 5 for what kinds of functions f x will these a exist and the expansion 5 be valid This question has an answer but this is not the place to go into detailsf The possibility of expansions of the form 5 obviously depends in a crucial way on the orthogonality property 2 of the Legendre polyno mials This is an instance of the following general phenomenon which is often encountered in the theory of special functions If a sequence of functions x 2x x de ned on an interval a lt x S b has the property that b 0 m n 1 q l 7 L ewcx 0 mm then the 4 are said to be orthogonal functions on this interval Just as above the general problem that arises in connection with a sequence of this kind is that of representing arbitrary functions f x by expansions of the form X fcrZmmox nl and a formal use of 7 suggests that the coefficients a ought to be given by s 1 lb efnwmeM fl Additional examples occur in Appendices B and D of Chapter 539 where 339I he answer we refer to often called the Legendre expansion theorem is easy to understand but its proof depends on many properties of the Legendre polynomials that we have not mentioned This theorem makes the following statement If both fx and f x have at most a nite number of jump discontinuities on the interval 1 S x S 1 and if fx and fx denote the limits offx from the left and from the right at a point x then the an exist and the Legendre series converges to 1 5Uofun for 1 ltx lt1 to f 1 at x 1 and to f1 at x 1 and in particular it converges to fx at every point of continuity See N N Lebedev Special Functions and Their Applications pp 5358 Prenticeallall Englewood Cliffs Nl 1965 tEIIFFERFHTlttLt eetuettatette tite ettttegetntettity with teepet te sttttetmet wetitght t fl1I39ttC391tifttTtS et tlte Herrttitte peI3rn etmtel5t end Cheb3r5ttetr pelyntemtelte is erte jy 39Tt1E H T39TIEdt The eetitsttteettety 5etutten eff thie AgreeAmp ef tebtzettte wee ene ef the trntzein te ettitetremente eVtquot pure m tthEm tTiE2t in the tntttteteeztttth end ee jt ttW39eItltEs39th eenturiee A150 Ch 39p lEt39 eenteEne e f iLl39tIf tut quotquottquotE3t39I39tquottEItTE ef the eieeeteet problem that UI1El39EFtiES ett ef these itieeetthet ef etpendtintg etltitehly tteett etett fvuiml tiDES in Fettriet eettee Least squares allpit im sii n Let fts be re tttttetten de ned en tthe titntentelt eel t p L end eenetder ttte prebitem eat epipmxittttettintg fftft tee eleeetg39 es pe5sitite in ttte seneet ef teeet eqtteree P pettynemitetie 0 mt degtee Estt P we ttttnlt ef ttte integral V t I rm PM at t as representing the sum ef the qtt l39ES If the deeietitens ef p39Jt39 trem flI then the prebtem J te minimize the trainee ef this itttegrel by e euttebt e etteiee ef pt It t39tJIt1tTls5 ettt tht tt the mittistttizing petynemitett ie pteeteely the eupmt ef the First it t terms set the Legen dret eettes 5 mt enPutI 0g ePtttIt where the eee ieiente ere gtvean by 6 Te preve thi t we use the feet that ell pettyntemieftts ef degree set are XpI S5 ihl e in tftte tetnt bgtPtt P bPntt The integral J een th f39E fEtt be Wtt39it tEt39t est I Jifx bkPx2dx I39ifJ J f39I 2 1391 p U 3 b E 2 33 fxFTtx six 1 e39e24quot39 1 1 te 39 73939 2 A dHbE 6 W bx egg d IE E J 39 0 2 H 0 2 1 395 at H 2t 1 2 ftt ex Jtgjl ftxfdx Ie E k Sinee the tee ete tted end the ere et em tltepetseL it o eteett tttet I asSil1IItES tte mteimumt tvatue when bk et fer t D m rt I I39ttt39 hypetheete retjutted by this etgurttentt is quotthat ft and ft392 must be integtrebTet If the fU5tT39l EttI Ct f 1 is et1 i eiettt Iy welt behetted te have 3 power sertes ettpensiern en the tntetvelt t 1 v 1 tthen meett etudettts eestmte that the bestquot pelynemtet etppt etxtmettietrte ten ftsxtj are given the partial ettm5 ef this fpewetr series The Iesulft we have e5ttehiieited here ehewe ttet this is false if eur ertterte n is eAppretttmetitten the sense ef least Sq39uatES some SPECIAL FUNCTIONS or MATHEMATICAL PHYSICS 347 PROBLEMS 1 Verify formula 4 2 Legendre s equation can also be written in the form d u x2gty391 nltn 1gty 0 so that 1 x2P39 mm 1P 0 s and d 2 C1 x P nn 1P 0 Use these two equations to give a proof of the rst part of formula 2 that does not depend on the speci c form of the Legendre polynomials Hint Multiply the rst equation by P and the second by P subtract and integrate from 1 to 1 3 If the generating relation given in Problems 1 and 2 of Section 44 is squared and integrated from x 1 to x 1 then the rst part of 2 implies that fg 3 P 2d2n l12xtt n 1 X xt Establish the second part of 2 by showing that the integral on the left has the value 39 E 2 Zn ni2n 1 i 4 Find the rst three terms of the Legendre series of 0 if 1 S x lt 0 a fx if 0 S x S 1 13 f X 6 5 Ifpx is a polynomial of degree n 2 1 such that I fxquotpxdx0 forltO1n 1 1 show that px cPx for some constant c 6 If Px is multiplied by the reciprocal r of the coef cient of xquot then the resulting polynomial rPx has leading coefficient 1 Show that this polyno mial has the following minimum property Among all polynomials of degree n with leading coef cient 1 rPx deviates least from zero on the interval 1 S x S 1 in the sense of least squares p 3J lFFEHEHquotFlA L EE1U5lTill34H5 pJ Q Q cli emnltlial gqualllmnl xEy l 1ltyquot 3 v plEy Cl P lwhere p is a nonanegative mnstanl is called Je55lels eqmlrllhn and its Earlfllfl1lti E lam kn wn ass Bane fquotum l Gn Tllhesel fufl liv llsu rs EIUSIB in Dlanllelll TBernolullllll 5 lrwesltlgati nln f Illa 5li ll3lti D 5 cznf a l1anglng cihain P 39UhlEH1 404 and appeared agariln in Etlll fm thwry of the lribratiuns ml al lsllrucular Il1El39l ll39 l Iquot IquotlxB and esl5El s slfuxdlles lf pllaneltary m liU139l Mme recerltly Elesszell fLl IE39 i I39lS have turned nut to have very diverse appllica Hang in phjyslcs amil engine rinlg in c nnlecl innl with lha pmplagatinn of wlaves eala5t39icit l fliuiid rmz tirn aftti espe cially in r1r39lanfy prolem5 P gmtentliall theory andl dl lusinn iIw ollving ryllinldric1ll sggrmmetryl They even UESEUT in same iI1terlEs1limg 39prulllI Erns Hf pure n1 IhET1E1IiiC5 WE pmalerll H few Eitpplli 7amp39U39ElIS in Apgplelnlldlix but zrst it is l39lEE39E v 5fL iEiquot39 la da ne th mare ifl ljPt7lll l i39 1 LI139E l esseall fU ClliUIlTlS alld nblaiml sme lof tlmir simpler pmperltlE5ll5l The e niliuun all ihelt39lfImnclim1llglxll We begin ur study of llw utllutimlg Elf ll by ntgicing that alfter ijiEi liquotiii1l lby 1392 tlhe CDEf lCilEIFII5 li f y and J are llflx and a 3 E a 51 P 1 and p 2 2 The urpihgin is 39lIhampl Ef l39E a rlegular sdnglllar p iinll the inldmiall eslqulatilaln 3045 is lquotquotlT 2 D3 antli the E p flEil l E395l are ms p and m It fulllows frmm lThel0rem 3lllA that Eq39Il39l 3IiD 1 has a s lnlltilml 4 Friaclri39wlh Wijl mlm HESHEI uf1ll4 lR lfl was lal lalmmls Gazrmarl 3lWllil39quotIl39lClLl39I39ll39Equot and an intimate l ri ln ul all lfiaus5 39willh wlmm h El r r 33p lndd fclr many ycars Hrs was title rst man lat LlE1mmil1E i3lE39 ilZ El4llquotETll39l quot the ll5tm39IaE Elf zlltd sitar lnzls parallav mtElrsllltn1irll llflf l fllil 1lllldrlrJ 3 i3t39al E fit Th l 1l39 fr Clgn39i il ll Iigl7llyedariss HT alramrt 3l IDquotT li 1EE lhe IEllEI l39lEll39 Elf lh EHFIWE Lturlflil In llM4 ht di5ml writszd lhal 5lri39ll llh brigh lrs11 star in lih illt3 has a llrawlliru g cmnT1p ln39i rm ll lllilldl is ll39llEr1El Dr wl1al ii nnw k w l as a brir1lalquot1r sflazrl 39lquotl1i CTnrrlal39lijnn DI Sliriu wlltl1 Elm rgize LE El planel but 39ll391E39 39rl1a5 ml 1 slat iancl E rl EqAMlwE Hr39 3 39l 1Ell I5ll F many lhljusaanzds ofquot Ifiml1 5 the drE l Il5il3l slE WalEl39 ls IiI1quotIE 3l lht um31 inlIl39E5lil1giilhj2l iIS lrl lhe zulllvclse H was IhE llrst l clE ad EllEr U3 bi disucwEreJ lili d lls39c1llll395 3 special p39l lilE iill lnntlarln lhlvalcalriltls Elf slvalllar lrwzillulllcjlns 5 Th anlirla liuhjualrzl il If llEdi am a VEIEJE amprra1l in p4v H Wms n A T rmrrlE n the Tlimry ljf Ba5lE39l Frmcllkorm Edi Eli l anwblridgcl Lll1iwritr lF39lrr5 nndrl n l9amp I4 T391iS is ill gzalgan tlLlan wnrrlt nf WE pages witlm 21 36Tpi3gr lvili39l gr lphl lflf 39IquotE39 l llzam5 Whlall lw 5l lalll xrclistiuslsl m 39ulnli5 tn liiHIE name than lha fm flh can 3 quotlE39l l39lll39lg Il39SELl39I Eff 5t39E Enli a EHKElr E l g mrEl39 rmzially 39lhilrEE r391IlL1rl r SOME SPECIAL FUNCTIONS or MATHEMATICAL PHYSICS 349 of the form y x Z axquot 2 axquotquot 2 where ao 5 0 and the power series 2 axquot converges for all x To nd this solution we write y Z01 panxquotquotquot quot and y 201 P 1n Pax quot2 These formulas enable us to express the terms on the left side of equation 1 in the form xzy 201 p 1n paxquotquot xy39 E n panxquotquotquot x2y Z an2xnp 192 2 p2anxquotquot If we add these series and equate to zero the coef cient of xquot 39 then after a little simpli cation we obtain the following recursion formula for the an n2P an i an2 3 or an 2 4 n2p n We know that a0 is nonzero and arbitrary Since a 0 4 tells us that a1 0 and repeated application of 4 yields the fact that a 0 for every odd subscript n The nonzero coef cients of our solution 2 are therefore 00 a4 02 00 42p 4 2 42p 22p 4 a4 a0 G 6392TTEj 2462p 22p 42p 6quotquot 350 ne1FFEHEm 1m EQUATIONS and me eelutim1 itself is I 14 39 0 quotp 1p 2 y Ip1 I I5 39 2 31Lv Map mm 3 39 p 3939 Aquot 23w P icy p r3939 5 The Bemei famerieen ef the m kTf d ef erder P deemed by Jpvx is de ned by puirtm an M 2Fp in 5 an that x ra Ape em Ax 2FZUgquot 1 21 nip 1 0I H mp n7 quot The meat useful Eeeeelle funetiens aree these ef errzler ii and L which are ex O 1 W HE 224 2242 339 73 and Jm 2 1nE1391 f r1in ee 1 393 1 1 x3 1 r 5quot i39 i a 33 Thueir graphs are shewn in Fig 53 These graphs display e ewrereE interesting preepeerties ef the functions JDxe end Jx each has 3 damped g er Vem 53 some SPECIAL FUNCTIONS or MATHEMATICAL PHYSICS 351 oscillatory behavior producing an in nite number of positive zeros and these zeros occur alternately in a manner suggesting the functions cosx and sin x This loose analogy is strengthened by the relation Jx J1x which we ask the reader to prove and apply in Problems 1 and 2 We hope the reader has noticed the following aw in this discussion that Jpx as defined by 6 is meaningless unless the nonnegative real number p is an integer since only in this case has any meaning been assigned to the factors p n in the denominators We next turn our attention to the problem of overcoming this difficulty The gamma function The purpose of this digression is to give a reasonable and useful meaning to p and more generally to p n for n 0 1 2 when the nonnegative real number p is not an integer We accomplish this by introducing the gamma function F p de ned by 13 Tp J t quot equotdt p gt O 9 0 The factor equot gt 0 so rapidly as I9 00 that this improper integral converges at the upper limit regardless of the value of p However at the lower limit we have equotquot39 gt 1 and the factor t gt 00 whenever p lt 1 The restriction that p must be positive is necessary in order to guarantee convergence at the lower limit It is easy to see that FP 1 PHP 10 for integration by parts yields b rp 1 M O tquote dt b b pf t e39dt O 0 b plimJ t 1equot39dt pFp b gt Q lim tpequot b gtoc since bl eb gt 0 as b gt 00 If we use the fact that m equot dt 1 11 0 then 10 yields F2 1F1 1 F3 2F2 2 1 F4 3F3 3 2 1 and in general rn 1 n 12 for any integer n 2 0 D 39JitF FtERE39MT39I M EDUATIDNS We betgant tf139 tjttsmssitun at the gamtma f39urt39Cti ll U d f the atssumptitzunt that p is ncnnnatsgmitvestt and we mentinyned at the C1L1tquotSE 1IZh l tha in tcgra1 V59 dues not existt if p quotHnwetvert we cant dE r1e lF p for matny negatttve pquots wittwutt tfhta aid of tLt1i5 integml if we write ft in tthe fmrn tLPtt139 This extemstimnt of thte dte nitttimn is necest5atfy for tthv pp39iiC Itit1t tS and it begim as fmltotwsz If 1 t p r G then El 4 p t It 50 the right siida at Eqlj lti 13 H313 a value and the left side of 13 is de ned tn have the VHIUE gitvan quottug 1hquotEv right sidet The tnezxt step is to notice thatt if E2 ti P H t therm 1 2 p 1 lt2 393 50 we can use 13 again EU d e 1quot p an the itntcwal M 2 1 in ttarms mf 15h vallgucs of Hwp 1 already tde tntcdt in the prt quottfiEIiL1 39t Emi It is clltear that this pmccss can be E nttnlt d ir1dE39 nitEly Furtthermore it it asjr to see tmtmt 11 that Fquot tfquot Hm Ftp Iim ti r aI39I fn al3939I 1quot pt 5 13 im awarding as p r B fmm 5 right or lefttt The futnctim Hp t3efh v es in a Simitar 39W Ly tntezar all negttivc iurquot1tegms and tVt39trEl Bfti11tTit j its grajpht 11135 tthte general tappetatance tsctmtwn in W S will taiuttltm new to know P t3 l lLl tDlIS fact that Tthtst is indicatt gd in the 39 gutr e and ittrs pmuf D terft tn the retadEr in Pmbaem Strata Iquot p nevuzztr 39u39ETt Tti5h EiS5 ths fttt1t3tiDfl 1f1F p wilt be cte ned and twat behtaved for an vtatluets at p it we agre that 1T p it far p D 1 H g idtiatis Enable MS 10 li n p by mMpm fur atl vaflttmes at p exxrzept nega tive integcrst antdt by fnrmuta M this ftuntztimwn has nits usual rnaaning when p is a ntnnegattive integ er Its LTE1Cip39Tf39DC Itfp 1 1quotp 1 is de nettt for alt ES and has the value I twthenetuter p is tat ZniE7g ti iquotE tinttegetquot v gaming function is an exttremety intttsertesting funetintn in its own trihghtt Hinwevet muquot purposte in inttr uduCitng it hem is sezattzty to glIa1l TlTtEEt ttlmt that funtti m JPr as da tn d s tmrmula r has 31 rneantng fur every p 2 L1 we tpttntt tlitlt that e v tn more hast bteetn Ehi39 VEd 5iI39ElGE Up H aw has at rneantintg farquot EtEF p 5 1 ft de nes ta pertf ect39t13s re5tpcttat39bttte funsctiant at 2 far all vatlu rs tlf pb Wilht tlt exxceptitgntt some SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS 353 39 39 I I I I I I I I I I I 39 I 39 I I 3quot l p I I 39 I I I I I 2 39 39 I I I I I I I I I I I I I I I IV I I I I 3I 2I II 172 I i 3 75 I I I I I I I I 1 I I I I I I I I I 39 39 20 I I 39 I I I I I I I I I I I 1 30 I I I I I I I I I I I I I I FIGURE 54 The general solution of BesseI s equation Our present position is this we have found a particular solution of 1 corresponding to the exponent m p namely Jx In order to nd the general solution we must now construct a second independent solution that is one that is not a constant multiple of Jx Any such solution is called a Bessel function of the second kind The natural procedure is to try the other exponent quot12 p But in doing so we expect to encounter dif culties whenever the difference ml m2 2p is zero or a positive integer that is whenever the nonnegative constant p is an integer or half an odd integer It turns out that the expected difficulties are serious only in the rst case We therefore begin by assuming that p is not an integer In this case we replace p by p in our previous treatment and it is easy to see that the discussion goes through almost without change The only exception is that 3 becomes n 2p na a2 0 and if it happens that p 12 then by letting n 1 we see that there is 354 eisssnm1TsLi Enusrrieiss ne eemipuilsinn tee cheese in y Hieweveri since sil we want is s psrtieuisur selutiien it is eertsiniy permissiie in put at The same preblein arises when p 32 and as 3 end se ewnr and we selve it by pugtting ei a3 II in sii cases Eiverything eise gees as before and we evbitsin s seeend seluiin anP Leixi The rst term ef this series is sltgtii Se JFx is nnbeundeid near I we Simeei J39px is beusnuled nes139J39 0 Tthese Ewe seiujtiens are independent and y esJFs39 e2fFs is net an in39Iteger is the general selujien Inf 1 The seluitien is entiieiy Eii er ent when p is sn integer 2 0 Fermnls 15 new beneeniesi 8f in l 3quotJquot2i39E ni m H 0 0 k r1ui m n i since the fseftisrs 139im H are series when n U 1 mi L n repiseing the durnmy variable rs 3 H m and CU39JTIpE 5 EiiiTlg by heginninig the summstien st in ii we ebtsin P e c iam cxi2 1quot ix 39 in c xf2 EFm iii in is im t d 1H is 1 iJmr This shew that JmJ is net inciepieni cienI efiis se in this case ElieX 0 y ElfrsiI is net the general sieiutieni emf ii and the siesireih C39Di39v39Mi iLiE5i At this partiini the sisery beeemesi rather eempliiestied end we sketeLh it very bsiie yi One pessibie spmscih is be use iZiTIE niethed emf Seeiien L which is essiily seen te yiieI dj MI f p V some SPECIAL FUNCTIONS or MATHEMATICAL PHYSICS 355 as a second solution independent of Jx It is customary however to proceed somewhat differently as follows When p is not an integer any function of the form 16 with C2 0 is a Bessel function of the second kind including Jx itself The standard Bessel function of the second kind is de ned by Jx cospar Jpx 1 sin pJr 7 Ypx This seemingly eccentric choice is made for good reasons which we describe in a moment First however the reader should notice that 16 can certainly be written in the equivalent form y cJx c2Yx p not an integer 18 We still have the problem of what to do when p is an integer m for 17 is meaningless in this case It turns out after detailed analysis that the function de ned by Yx lira Ypx 19 exists and is a Bessel function of the second kind and it follows that y cJx c2Yx 20 is the general solution of Bessel s equation in all cases whether p is an integer or not The graph of Kx is shown by the dashed curve in Fig 53 This graph illustrates the important fact that for every p 2 O the function Yx is unbounded near the origin Accordingly if we are interested only in solutions of Bessel s equation that are bounded near x O and this is often the case in the applications then we must take C2 0 in 20 Now for the promised explanation of the surprising form of 17 We have pointed out that there are many ways of de ning Bessel functions of the second kind The de nitions 17 and 19 are particu larly convenient for two reasons First the form of 17 makes it fairly easy to show that the limit 19 exists see Problem 9 And second these de nitions imply that the behavior of Yx for large values of x is matched in a natural way to the behavior of Jpx To understand what is meant by this statement we recall from Problem 243 that introducing a new dependent variable ux g yx transforms Bessel s equation 1 into 1 4 2 uquot 1 4x u 0 21 When x is very large equation 21 closely approximates the familiar differential equation itquot u 0 which has independent solutions 356 DIFFKEREHTLIAL EquqaT1sn r5 ult com and u21 sin 2 W3 39Izl1ere4fm39e expect that for lalfge values sf 1 any Bessel functim1 yfxj wtill b hava like some linear mmfbinatinn if 1 1 Tr ms I and sin 1 This expectation is supported by tha fact that at pn j i r F E E 3331 In and 2 Sim f V H V n pi j Mr ymxj 0 Ex h 4 E4 1 quot39 39 A 35 I 4 Wham rxJ and r2x am bUundEd as 1 is rin PROBLEMS 0 Use S and 3 Cr Shaw t1391at w mn4mm N 939J13 quotTquot 139939 aA39x T USE Pnzrublem 1 and RCnME39S th rem ti sheW tEl 11t 13 Bmrwsacn any twU p sitiwi E6I U m1fJ J them is 3 E6113 Off MI b BE39E wEcn any Wm p rrsi tive zems of mix there is 3 zero of x m mrding to the dr3fimitimn 9L 1 I i J I39equotcf 3 Shaw mm the change of variame 53 leaE13 In 1 39 3 r 2 a 2 L E h Sztince 339 in 3 quotis a dumn1y vaj iaIble WE mm write rltgt 3th Iii Jr Etig1391 l H I F SEE Warsaw up m39I chap Al39i nnrt r4ec te 5 ml C nwL139rani and Hi bert Mmfimds Hf Ma fmmn iimE39 FIja393fiE uni 1 pp 3334 526 lntercianctWiI y39 Haw quotTnrk 1953 some SPECIAL FUNCTIONS or MATHEMATICAL PHYSICS 357 By changing this double integral to polar coordinates show that 1 2 Il392 00 2 I 4 J equotrdrd6 Jr 2 0 0 so 4 Since p F p 1 whenever p is not a negative integer 14 says that l H Calculate and 31 More generally show that H 2n 1 J 2 22nln and quot quot uQ 3 391It for any nonnegative integer n 5 When p 12 equation 21 shows that the general solution of Bessel s equation is expressible in either of the equivalent forms 2 V1ct cosx C2 sin x and y cJ2xp c2J2x It therefore must be true that J2x a cosx b sinx and cJ2x ccosx dsinx for certain constants a b c and d By evaluating these constants show that 2 2 J2x Esmx and J2x cos x 6 Establish the formulas in Problem 5 by direct manipulation of the series expansions of J2 and J 2x 7 Many differential equations are really Bessel s equation in disguised form and are therefore solvable by means of Bessel functions For example let Bessel s equation be written as dzw dw 2 2 2 z z w 0 Z dzz dz p and show that the change of variables de ned by z ax and w 2 yx where a b and c are constants transforms it into 2 d xzg 2I 26 Ux azbzxzb C2 p2b2y 0 Write the general solution of this equation in terms of Bessel functions SOME SPECIAL FUNCTIONS OF MATHEMATICAL PHYSlCS If 4 and 5 are first added and then subtracted the results are 22x Jix Jplx 6 and x J1lx plx These formulas enable us to express Bessel functions and their deriva tives in terms of other Bessel functions An interesting application of 7 begins with the formulas 2 2 J2x Jgsinx and J2x J rcosx which were established in Problem 465 It now follows from 7 that 1 2 12xgt J2xgt 3 5 cosx X JTX X and 3 2 3 39 3 J52x J32x J2x S39fquot 3 H Sm X JEX X X Also 1 2 132x 2 quot112x J12x quot Cosx 51111 X 739L39X X and 3 2 3co 3 39 152x 32 1120 zsx Smx quot C051 rrx x x It is clear that calculations of this kind can be continued inde nitely and therefore every Bessel function J12x where m is an integer is elementary It has been proved by Liouville that these are the only cases in which JP x is elementary7 Another application of formula 7 is given at the end of Appendix C where we show how it yields Lambert s continued fraction for tan x This continued fraction is of great historical interest for it led to the rst proof of the fact that J is not a rational number I 7The details of this remarkable achievement can be found in Watson op cit chap IV and in J F Ritt Integration in Finite Terms Columbia University Press New York 1948 The functions J2x are often called spherical Bessel functions because they arise in solving the wave equation in spherical coordinates 36 D tFiF39EitEHTiAlLt ee usittieiss Whert the d39iHVEI39 ttii tiw fetrttutiss E attti 3 ME 39W FiiE in ithe term J ix39FJP 1F39tT dis 2 xiiip 2 2 Iat IIZi gt 0 JxquotPJMx 0 ts t xV e PA then they serve fer the 39il1ItBgI3931il39C3939I39Ii ef mang simple E3tpIquotES5iDFtS eertiteittitng Bessel fhnetiefntstt Fer esempltiet when p E 0P 8 yields J XJMI ds 2 9 ti in the ease ef there eempiieeted ignitegreifst where the espeinent ties net meteh the fiI tiiE 139 ef the Hesse fuhtttieh es it ees in 8 1quottdi 94 i li gft ilii that parts is usttsil3r tneeesssty as s SL1pp1tBmE vE3I teei Zerttslt ends Bessel series It fetitltewst frem P1quotEiiJ4i1Ern 24 3 that fer E vi39EIquotquot39 same ef p the furtetieh tJpsV has an i tfli i E ntttner hf p e sitive sets This is true in ptsttiteeiet ef i s The sets hf this htttetiett ste tkhewtt he at high degree est seeutsey end their teiues are given in iT1 I1jgquot vselumes f39 mathietmstiieeii tshies The fhti ties are sppreximsteigr 24043 3653 117915 enrtit 149 3U9 their stteeessive di u f l v tare 31153 0 seed The EItit Esp diIiIg pesitise zeros End t Vi i gEtIquottE lI39I139tTquot ES fer hts39 are 3831 Tn 5Equotj iUJir3935 i33f23 7 and 164TU6 end 0Y 315 T9i and p 39Netieie here these dihietieneest cetn rm the gtIetentees gitvien in Prnhlem 251 What is the fptlt39pS E eif this E cefn with the sertjs et Jwt4t It is e etn tteeessety tin msttthemstiiesit ph3tsies the etpantdt s given fuhetiert in ii1Et mS set Bessel funetitmst whete the particular type set E itlp 39nEiDi39t d p ti ls en the pt 39blE39m est hand The sitnpiest tend rnest ttsefui esipsnsiens evf this kind are SE5i39E5 ef the term eJt aJtlix sEJt sE p E A 11 n where ft is dtet39inedt cm the inteirsei P3 Es i sends the u are the tpesiijtise seres ef seme iiseti essel fu nl Iiim I1 iris with p z w y have ehesen the itnitetsel U 5 I p 1 tettths fer the sake DESimpI iC1iIt39 and all the fet tt1uies ggisierh eew seen he esdepted thy he sifrttpiet ehsne ef vsrieibie Us the ease heitquot s ftmtetietn ei heti ens sh ihtttersait atquot the term m E s E at The reie ei S39itClI t etpsintstiettts in Vpthysiesil ptehIehts is sirntilistr te thst hf L gB1 IEitquottt series as iiiIii FilSCi1quot 39lIZiEdi ih zhpviptenrditt 3 where the tehie m eettsidered iinselses tetrttptetaturies j a sphere Itt Appt dihi B we rtiemenitsttirste the d of 141 in seising the twe tdiitt1enstitttttsI wave erzluetien fer Ob vits39tihg Eit39t239Ui I39 emettthrentet some SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS 361 In the light of our previous experience with Legendre series we expect the determination of the coef cients in 11 to depend on certain integral properties of the functions JIx What we need here is the fact that 1 0 if m 5 n J Am J 1 d 1 12 ox p X p x x 2J1l2 ifm n In terms of the ideas introduced in Section 43 these formulas say that the functions JpAx are orthogonal with respect to the weight function x on the interval 0 5 x S 1 We shall prove them at the end of this section but rst we demonstrate their use If an expansion of the form 11 is assumed to be possible then multiplying through by xJplx formally integrating term by term from 0 to 1 and using 12 yields I I xfxJpAmxdx J1ltAr 0 and on replacing m by n we obtain the following formula for a 2 J on xfxJ lx dx 13 JA2 P The series 11 with its coefficients calculated by 13 is called the Bessel series or sometimes the FourierBessel series of the function f x As usual we state without proof a rather deep theorem that gives conditions under which this series actually converges and has the sum f x Theorem A Bessel expansion theorem Assume that f x and f x have at most a nite number of jump discontinuities on the interval 0 S x S 1 f0 lt x lt 1 then the Bessel series 11 converges to fx when x is a point of continuity of this function and converges to fx fx when x is a point of discontinuity It is natural to wonder what happens at the endpoints of the interval At x 1 the series converges to zero regardless of the nature of the function because every Jl is zero The series also converges at x 0 to zero ifp gt 0 and to f0 ifp 0 As an illustration we compute the Bessel series of the function fx 1 for the interval 0 S x S 1 in terms of the functions JAx where it is understood thatthe 1 are the positive zeros of Jx In this For the proof see Watson op cit chap XVIII Dnt rltEREmTmL mu39aTI0Ms Ei s L is JIIJ 1 xaL39x an E 0 HM J n3l 2 H E E3 MEI we see that L J L J xJm ix n Ax quot1511 U in ED i H a M It fU UW5 that L W VAR M gm 7IAquotl is the dasired Bessel s eIies LPVrmuf5 f the h g Ii39II39 prmp Ertie5 Tm estahli5h 12 we beqgjiqn with has fiatquot t39haI y JJxI El sDlutAi0n miquot p PE c c 0 J I x3Jr If H and 3 are distinct p1wsiIi39wB DnsLant5 f ll wi IfI1a1 the funclim n5 ups Jpnx and 1r7c 8 JbxI satisfy the eqjVuaiiiwn5 M M H2 3 U 1 r2 and u39 E E 5 D 15 1 J We may mu4MEply H1e5e equaIVi ms by 1 and u the sumract the results In r py r r 2 2 g 1 S L M u 1 E U 42 b ii jam and after mu tip1ication this baeconma ExL r U U 9 i 1uu 16 Wh1 16 is imegratad fmm x 2 U U 1 E 1 WE gal 1 E13 cIEJ xuudx xu39v u u quot5 H SOME SPECIAL FUNCTIONS or MATHEMATICAL PHYSICS 363 The expression in brackets clearly vanishes at x 0 and at the other end of the interval we have u1 Ja and 111 Jpb It therefore follows that the integral on the left is zero if a and b are distinct positive zeros Am and 1 of Jx that is we have 1 xJlxJlx dx O 17 which is the rst part of 12 Our nal task is to evaluate the integral in 17 when m n If 14 is multiplied by 2x2u39 it becomes 2x2u u 2xu392 Zazxzuu Zpzuu 0 or d d d x2u 2 3 azxzuz Zazxuz 6 2p2u2 0 so on integrating from x 0 to x 1 we obtain 1 212 xuz dx x2u 2 azxz p2tu21 18 0 When x 0 the expression in brackets vanishes and since u 1 aJ a 18 yields 1 2 1 r 2 1 P2 2 J0xJax dx iJpa 1 Z13Ja We now put a 1 and get 1 2 1 I 2 1 2 xJpLx dx Jp11 JpL 0 2 2 where the last step makes use of 5 and the proof of 12 is complete PROBLEMS 1 Verify formula 3 2 Prove that the positive zeros of Jx and Jx occur alternately in the sense that between each pair of consecutive positive zeros of either there is exactly one zero of the other 3 Express J2x J3x and J4x in terms of Jx and Jx 4 Iffx is de ned by Nquotquot 39ltgt39lt C ll l H NI R l 2 fx gnu gt s o DHr EFIHTIai EQLJMIQNE 5hCrw that fmzixmm rave 39lrquotquotrwE A39rai Ii it where the A ar the pm5iEiquotvE I39EI 5 E139fJux39 0 t If x J fur tIh E intEWal q Y r N shaw hast its Bessel snriies in j funltvt39ia3ns J dlQL7 E where the AH are thc porsitwe TEEEFGS Elf JLe1 J Qj Qj q I Qj 3395Ip 1quotA39P I JFlHx p USE thin 11UIHMEJH Df FrnJ hEem pom in Sfh W mrmalliyr that if gtfgc iiis l w fI fbitahavr d fquotllI7 lEIi iIEII I7quotI an the interval D E 1quot m 1 than 0 d 41 Igv xjl n e l v xg rJ UI I39 By t king gquot I and x uziedixsIsa that J 4 I 1 p b 1 1 J j Lid ii T W I 4 0 1 p E 1 In i i 1 161 p lf p V 2 The p itiiere 11rem5 mi 5int ara IE 21 335 Mb f U54 lzht mE539i1Il nf Fr b em PIb and Pr whlEm 465 in Show that H2 Vquot 11 9 Mquot 6 nil and Ob Shaw that ma cVhang e turf depEnrdintA ramquoti1blc Ell d p quot By 3 mg tr39an5fUr mr E111 spacial RriL39cati Eq 111tinJ1 Ty 391 0 K Q pm m dx Jquot I in if dlu quotTquot B39Z39xquotquot u ID P2 If c 5 2 UIEE Prznbl m 46 ti 39lta391ww that this Eeq IJElilT 39Il 1 i5 sUv39abe in terms Bf taIe39mertIatry funtimns if and nijy if U 4kf2k 1 far 5Dmcr itruteger P Wham m 2 the 311b539Iitutitun R Mfr 39Eramsfm39m5 i mfMi 5 aquarium mm an equation wM1 539Ei39HsT39ElbIE wmri1a hle5 t h1tA lfmas an eiemunta rja u 11timnV some SPECIAL FUNCTIONS or MATHEMATICAL PHYSICS 365 9 Show that the general solution of E X2 yz can be written as J 346V2 CJ34x2 CJ i4x2 J14X2 yx APPENDIX A LEGENDRE POLYNOMIALS AND POTENTIAL THEORY If a number of particles of masses m m2 m attracting according to the inverse square law of gravitation are placed at points P1 P2 P then the potential due to these particles at any point P that is the work done against their attractive forces in moving a unit mass from P to an in nite distance is Gm Gm Gm U 0 A quot 1 PP PP2 PP where G is the gravitational constant9 If the points P P1 P2 P have rectangular coordinates xyz xyz x2y2z2 xyz so that PP x x1gt2 y iygt2 2 21 with similar expressions for the other distances then it is easy to verify by partial differentiation that the potential U satis es Laplace s equation 82U 82U 82U 7 quot7 quot7 0 2 8x 8y 8x This partial differential equation does not involve either the particular masses or the coordinates of the points at which they are located so it is satis ed by the potential produced in empty space by an arbitrary discrete or continuous distribution of particles It is often written in the form VZU 0 3 where the symbol V2 del squared is simply a concise notation for the differential operator 32 82 82 t 18x2 ayz 822 9 See equation 2117 te tttaEaat4TaumL aertta39TIett5 The ftmetitn 0 he called a gr U itatEE0rt 1t ptJ39terIt ttt If we wetk instead with ehatiged particles at charges an pl gm thert their etheetreatatth petentfiet has the same term as 1 with the mquotIe repltaeerd by and Gquot hy Cetttenthh eehsttah ttt SD it atlse eatiahee Laptaee 5 equtatietnt This equatttie n has such a wiade variety ef atpptteatietnat that he etttua P a branch hf analyaitst in its ewe tightt hhnewn as p fetIft7t theexry the t EIaIEd eqttatttian 2 called the hem eqaattenj eeeare in pre39tateme ef heat eeadtuetieni where U ie newt a fu ti ef the time I as well as the space Dtt dimitES The thave Eqttttlfti tt atatut 4 ate We e W 5 ie C ttEEtE39Ei with ttthffatCtrj quot phhetnetmerta We add at feta hriet E tm lEt39tti5 ah the phy Si3Celtl theanhtg at Equotq 3tli 5 3 anti p tEquatttieh is the thttteedimeneiettnat eeunterpatt P the eneadimeneiena1 WHVIE Eaqtt tt tUjg which we have already dtiaeuseed utte tttaltly First Latptaaetete equatiena 3 tnahea the same eert ef etattemeat ahettt the ftmettiett Uh ea the enedtirneaaetaelt eqttatten d yfaitl U maheta aheutt at fLmtiet1 tt ef the single 1r39a1 i 1l Sv 139t U the Ll5attter eqaatiaan imptitee that quotJ haa the linear fettin Eh me and Eh39E13939 such f39t139ZnCItDn has the ptepertVy that its tvatue at the eetttet ef an inte rvta equafts the quotItl39iE3gE ef its values at the 39Et tdp 39it tt5 It is clear item 1 that eehtttene at LEtpi 339Eij 39S equtattteth need at he Iineaart ftlinC ti xtt5 ef x y and 2 in feet they can he very eemptieattted tirttteetd Nevettheteae it ean be proved aha was dttaemetred hay Gauaa that any aehttien at htaa the atBTW r emarhahh1e ptquotD pvEtjrquot that its tt llglE at the eettter ef sphere equate the a39verage ef its trainee en the eutrfaee hf that ehteretm Mate genetat1y the hthetiean Vii eazn he theL1ght ef as a reugh ttteaeure Df th dt hetzentee laetwetetn the avetrage tramet ate ps he the surface at a etgnah sphere and its exact quotIr Eh39ll Et at the center fm etamele if U represents the 39I1EF pBl aTu139E at an at bttraty paint P in he solid p amt WLF peeitittre at a eettain paint 0 tlhejn the atue ef U at P15 is itt fgenerat meter that its values at htearby paints We 39thetefete Etttpti z t heat he flew teward Pin t iSiI1g the ttemjper tatute there and since the tempEr tL1rEt U is t3rUhf t piDS1iE i tfE at p This is eeaenttiattyt what the heat equatiea 4 sage that pfUtpDt Tt i nei te WU and thee the 5att 1e sign If the I39E39mpEI tUI7E U reaehes a ettetatdy state th139Ltug39hDut the W The twteedirneheiettat vtefai at that pte pettry ia gittett in Ptehlenzt 421 SOME SPEClAL FUNCTIONS OF MATHEMATICAL PHYSICS Z1 lt1 x FIGURE 55 body so that 8USt 0 at all points then VZU 0 and we are back to the case of Laplace s equation We shall have occasion to use the formulas for VZU in cylindrical coordinates r6z and spherical coordinates p 942 as shown in Fig 55 These coordinates are related to rectangular coordinates by the equa ons xrcos6 yrsin6 zz and xpsin cos6 ypsin4gtsin6 zpcosltb By tedious but straightforward calculations one can show that in cylindrical coordinates 82U 1c9U 1 82U c92U V2 395f52quot5a327 6 and in spherical coordinates 7 1a2aU 1 a V2U H p 8p p2s1nqb8 av 1 a2U 2 0 8p Sm 391 36 p2 sin2 4 53539 All students of mathematics or physics should carry out the necessary calculations at least once in their lives but perhaps once is enough Steadystate temperatures in a sphere Our purpose in this example is to M DIFFZERENTIAL EEIUATIDNE iltuetterete as eimply es peesmle the mile ef Legerndre pelynemiele in tsetlwring eertaim beundary vame prebltemts eat mathematieel phystee Let a seii sphere etf r39adiue 1 be platted in a spherieel eeerdinete S139tSs39139iE1 39I l with its eenter at the erigin Let the euI f aee be held at at specified tetfmptetrature f which test assumed Ate he mde pend ent tef 6 for the eatktet e f simplicity quotuntil the ew ef heat pmdtttcee a SttE d State fer the temperatute Tpqt within the ephseree The tprethlem is te nd an explieit retpresenettatient fer the tempseraetutre funetitetn Tp4qh The stetedystate temperetureet T satis es Laptaee Ls equation in spherieel eeetrdinates and einee T ees net depend en 0 7 ellews us te write this equtetitent in the term tag 13quot 1 3 S30 tp Ep A sine Bee 1 ae um 4 Te salve 8 eutnjeet test the gitretn beunderr veUndiit iewLn W11 my 9 we use the methed ef Seprtl n elf veriebiesg that its we seek a eeluttiten E et 8 ef the fermt Te upuq When this e xprgtessien is insertedi in 8 and the veriatbIes are eeperatedtw we ebtairn i2 1 i 3 13 I Km 5 dp39 vein M Squot 39d The erueitail step in the methd its the fellewing ebstervetien since the left side ef equatien MI is 1 fUI1CtLi39D39IT ef p alente and the right eide is at funtetien If e alene reetelt side tmust be eenetantet If thie Ie1391eta nt taIled the teeperetiete zeemmnt i5 deneted by 91 then 10 splits into the twe erttdirnargy di erentia eque ti0ne dam in z quot in and It d O B I g l t AU Equatien 11 is an Euler eqeettien with p 2 and q 21 50 its indietiel equetient its mIm 1t2m tU U1 m2m Jt1 The expenentts ere thereferte l M 41 and the generate rselttuteien H Martyr IrnIiems of greater eempljettityr are diieeueeett in Leljetlettquot apt exit elttep SOME SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS D of 11 is u Clp l2Al4 621012 A14 13 01 u c3p 2 c4pquotquot2 log p To guarantee that u is singlevalued andbounded near p 0 we discard thesecond possibility altogether and in 13 put c2 0 and 71 3 n where n is a nonnegative integer It follows that 1 nn 1 so 13 reduces to U Chiquot e and 12 becomes d2vcos dv 1 0 n n v dgbz sin qb dip If the independent variable is changed from qb to x 39 cos qb then this equation is transformed into dzu dv Zx nn 1v 0 15 1 2 xdx2 d which is precisely Legendre s equation By the physics of the problem the function u must be bounded for O lt 5 3 Jr or equivalently for 1 S x lt 1 and we know from Section 44 that the only solutions of 15 with this property are constant multiples of the Legendre polynomials Px If this result is combined with 14 then it follows that for each n O 1 2 we have particular solutions of 8 of the form anp I C0S ltigt 16 where the a are arbitrary constants We cannot hope to satisfy the boundary condition 9 by using these solutions individually However Laplace s equation is linear and sums of solutions are also solutions so it is natural to put the particular solutions 16 together into an in nite series and hope that Tpqb can be expressed in the form as Tplt1gt Z apquotPnC0s 39 17 n The boundary condition 9 now requires that flt1gt Z anPnC0S n or equivalently that DC fc0s x Z aPltxgt lt18 n0 370 1t139tiEeeit rLeL eeiueftte e We knew fmrn Seetiee 45 that if the fugmtin fee39i Jc ie eu iie39ietr1tt1j3ri welt rby iml iif 5 then it een he eitpaeded inite e L egeectire eerieei ef the ferm 18 where the eee iieiente i ere giv39eti by it rt O X e eequotquot xF r c 19 these C i gi ili fi 1 is the desired eemitien ef em preblere W have ieund the eetutiein ii739i by tether tetmei pI EBdui39eES and it SJh IuM be Vpeinted rent that there are d i JE lil1 E queettiene efi pure meth e nietiee iirwetlmquoted here that we Tl39iev e met E l3ItihEd en at ell Te at Aphiyeieieit it me3r seem eiviie ee tthet e eeiid tietiiy wheee eurfaiee 39EiEmpE 1 HI1LIJFE is speci ed wit ieetiiuetiiy attain e de nite emit utquotniqiue e39t etetdi39 stat e ternpera iture at every in1etiefr peinl quotbet methiemeittieierts ere enhemiley aware that the ebeieue is eften tfetsefg The teecalled Dritiiehleti prevb l em pevtentiell tthieeltry lquot39ElL1iiI ES e Tig TD11S pteef ef the exieitene e anti ueiqiueimee ef e J t ETl1 t l funetimn threughiee l e riegiiee that aesttmee giv lll values en the b DLl da39I t This problem was eeived Pn the eerhr twentieth CEI391 ilquot quot the greet Germetin metheimetieien Ittiibetrti ieir very geneirtel but preeieely dLe ned typ ee ef bieuedteritee aI391Ld bieuinderhy ftmettieee The eie eitiritnetatiie dipee 1IMElIi139Is i 39L The geinierertigng reletien 1 ea 4134 2 my fee the Legendre pelynremieie is tiieeu5eedt in Pttebiemis 441 Y and k As a direct piiyeiieat iiliuetretiein ef its v1eli1ie we ueet it Ate nd the peitentiei due tie twe P iI39l I ieihrergets ef equal mE39lgni 1IM39dEi but fptPGSitE sign 0 these ehargee are placed in e pveiar ee erdiinetei reyisitiiettmt A56 IF URE 56 Senfie feitigr eimple exemplee in 39W itiEh the etatemee t jest ft lFiCiE is 39ei395eA ere gieen in W D tietljtegg Feune39etieet ef tP39etee ie Wiser p 2351 Spritlger New ftzutl 1929 Erieetein El greet melter ei epherierns eeid T he rereet end meet eelueible evt eii it1teltleetuel treite is the eepeteiity tie de ebt the ehtrie39uet quot SOME SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS 0 then with suitable units of measurement the potential at P is 51 6 U 21 1 r2 where r r2 a2 2ar cos 6 and r2 r2 a 2 2ar cos 6 by the law of cosines When r gt a we can use 20 to write 33 1 1 Plt MCY r E r1 2cos6ar ar2 r0 quot C03 r 39 and similarly 1 1 1 1 quot P cos 6E r r E F1 2cos 6ar ar2 quot Formula 21 can now be written F1 U S Pcos 6 P cos 93quot 22 n We know that the nth Legendre polynomial Px is even if n is even and odd if n is odd The bracketed expression therefore equals 0 or 2Pc0s 6 according as n is even or odd and 22 becomes 2 9 2nl U 72 2 P2n1C0S 6 n 2 rqPcos a 39 Pcos 9 3 23 If we now assume that all terms except the first can be neglected when r is large compared with a and recall that P x x then 23 yields cos 6 r2 U 2aq This is the approximation used by physicists for the dipole potential APPENDIX B BESSEL FUNCTIONS AND THE VIBRATING MEMBRANE One of the simplest physical applications of Bessel functions occurs in Euler s theory of the vibrations of a circular membrane In this context a membrane is understood to be a uniform thin sheet of exible material pulled taut into a state of uniform tension and clamped along a given closed curve in the xyplane When this membrane is slightly displaced from its equilibrium position and then released the restoring forces due iDlF FFElEH39l ils L EQUATIUHS the the desfennetih cause it to isihratle Chit preihlem is the eniislyze this viihretiaelhisl metiwn The 39LEIiIilsI E lMl ell motion Muir d lSCll39SSi E1i is siimilsr in that given in Seeirien M fer the vihirmiiin lsiiringig that we rnelse selverel Sll11 illfp ii g sslsump times that enable us in ifermulate e pertiel dii IElTEli iliil il equelti in and we hiepe that this ieqiuetiien deselfibes the mmien wilhi e rreasieneblle degree of reeeurieley These assump1ieins eein he sfummerieed in s singliie silstem emi we eensider nly smell eseiilletirms of as freely vibrating memhrsne The lvsriillelus ways in which this is used will appear as we preeeed First we essiume tlhst Q vihr39stiienls are sew smell tihat eec h peint of the mlemlmiene memes enly in the es ldiirleeitien withl irxiispleeement set times I given by serne fenletieni z zxyr We eensider 3 small piece ef the membratne iF igii 57 bounded z seirtiesl pliiahesi thmugh the felilewing pD7il1tS the xyplieinezl Lye ix early x ampxy eiyi and xy Ely If PM is the eehsvlent mass per unit area then the mess ef this pieee is M P y and ilmy Newieni39s seemed law elf metien we see that 32 F p x U is the terse eeting p in z idireClieni Wihien the memhrerle is in its equililwium 3DSil l l i me eensiasmt tensieh T ihes the fellewing P h1f39LSiiall meehin Aiming any line segment ei length Ase the meIterlisl en seine side ieserts 3 force nermiali the the segmiemz and mi magnit39ulie T en the metei ial in the ether side in this case the fences en eppesi te edges ell mlr smsl pieces are parallel he the xyvplenie and esheel ease 4ImiD EhiEra when the membranes is e uIquotv ed as in the fresen instanlt r1f metien shown in Fig we assume ilhiet the deferlmiatiien is so small that the tensien is still T but new sets perellileli the the i lflgl I lii plane and thlerefere has an aplpreeiiahle vertiesl eempenenti It is the euwaiituire Fl G Ll W M some SPECIAL FUNCTIONS or MATHEMATICAL PHYSICS 373 of our piece which produces different magnitudes for these vertical components on opposite edges and this in turn is the source of the restoring forces that cause the motion We analyze these forces by assuming that the piece of the membrane denoted by ABCD is only slightly tilted This makes it possible to replace the sines of certain small angles by their tangents as follows Along the edges DC and AB the forces are perpendicular to the xaxis and almost parallel to the yaxis with small 2 components approximately equal to a T Ax5 and T Ax59 z 3 yAy 3 y so their sum is approximately 8 8 wlti lt igt l 3 yAy aye y The subscripts on these partial derivatives indicate their values at the points x y Ay and x y By working in the same way on the edges BC and AD we nd that the total force in the z direction neglecting all external forces is approximately FWl lw gt so 1 can be written TaZaxxAx 39quotquot 82axx TaZayyAy quot39 82ayy m 2Z Ax Ay c9t239 If we now put a2 Tm and let Ax gt 0 and Ay gt 0 this becomes 282z 822 822 2 55yquot5 E which is the twodimensional wave equation Students may be somewhat skeptical about the argument leading to equation 2 If so they have plenty of company for the question of what constitutes a satisfactory derivation of the differential equation describing a given physical system is never easy and is particularly baffling in the case of the wave equation To give a more re ned treatment of the limits involved would get us nowhere since the membrane is ultimately atomic and not continuous at all Perhaps the most reasonable attitude is to accept our discussion as a plausibility argument that suggests the wave equation as a mathematical model We can then adopt this equation as an axiom of rational llmechanics describing an ideal membrane whose 374 ifzF FE FI EHTIM e eumTev1eee me 39chemetieel beeeewieere megr er may met match the aetuel beeheawieer not real meemehrenee 3 The circular meemheremee new specialize to nzhge eeee ef e EiilT El af meeemebzrenee in whiegeeh it I ineturel te use paeiler eeerdinamee with me erigim Ieel ed at the LEEkI1L4tE f I1 Fermme ef Appzemilix A shows that in lihtie ease the wave Eql at1 IU 39 takes the ferm 2 823 13 1 ze 32 an e gt e g HP p gt xquotr lt Sr re Er rel 51 EH where 2 V zfr 51 is e fiunestien of the gpefler eeerdlinetee and the timeee Fer eeenveneeieeenee we esieumee that thee emembraene has radius 1 end is theraefere clamped te its plane ef eequ ieb eeum along the eierele r w Aeeerinegee1ye eur b eum1der3r eieeenditiee 633 U 4 The prre bIeem Ate d p v seflueizien ef 3 that eetie es fthie boundary eeeencmiene eende eeer39ta ien LiJ iIIiamp1 C D diIi De iS te be speci ed eleter In eepplyieg the er39tende1 d meethed ef eepaxratien ef veriee bleese we bein with e eearrl1 for pemeruler s e euetiens ef the form ezn9 M r u 9w139 5 when 5 p j inserted in 3 end thee realm is reeerremed we get Mr j r em Tf llif 339 ee Wm Since thee left side ef eque ee a is a funeeieeen only ef r 0 a and the right side is e fueetiene eenly of mth sides must equeel e eenetente eF1er the membrene In vibrefte w f emuet be periedie and the right side ref 6 ehews that in erdler me gueereenetee rthie the separetiree ceenetem must be nzegetirvee We her eifer e equate eerzht eidee ef 6 I 413 wilh A J and ebtein the We equeewt ieeene wequot39r lzezwe xeje U 7 endi X X U Mfr r INF r pkU U 3 111 the L uee1ieen quot39WIhren e reEiene reeehaeniee39 we reeemmemi the eiiHumei39neEeing remerl gs ef C Terl E39SeI J E7I H in the H niTf Fy ef Me e39heeE39es may 334 3 lD SpIringen New i e139k 19938 SOME SPECIAL FUNCTIONS or MATHEMATICAL PHYSICS 375 It is clear that 7 has i wt C1 cos lat C2 sin lat 9 as its general solution and 8 can be written as 1 2511 39 2239iquot rurrurlr Uw In 10 we have a function of r on the left and a function of 9 on the right so again both sides must equal a constant We now recall that the polargangle 6 of a point in the plane is determined only up to an integral multiple of 21 and by the nature of our problem the value of v at any point must be independent of the value of 9 used to describe that point This requires that v must be either a constant or else nonconstant and periodic with period 271 An inspection of the right side of equation 10 shows that these possibilities are covered by writing the separation 10 constant in the form n2 where n 0 1 2 and then 10 splits into vquot6 n2v6 O 11 and r2uquotr ru r lzrz n2ur 0 12 By recalling that v is either a constant or else nonconstant and periodic with period 21 we see that 11 implies that 116 d cos n6 dz sin r16 13 for each n regardless of the fact that 13 is not the general solution of 11 when n 0 Next it is clear from Problem 467 that 12 is a slightly disguised form of Bessel s equation of order n with a bounded solution JIr and an independent unbounded solution Ylr Since ur is necessarily bounded near r 0 we discard the second solution and write ur kJAr 14 The boundary condition 4 can now be satis ed by requiring that u1 0 or J1 0 15 Thus the permissible values of A are the positive zeros of the function Jx and we know from Section 47 that Jx has an in nite number of such zeros We therefore conclude that the particular solutions 5 yielded by this analysis are constant multiples of the doubly in nite array of functions Jlrd1 col 19 dz sin n9c1 cos lat C2 sin lat 16 where n 0 12 and for each n the corresponding Vs are the positive roots of 15 3T6 I393IFFERENTML ECtiiquot TlEtt i5 iSpeeial illfilia yl 39 i lt The ehrwe dieet15eien is intend1ed the ehiew how eeeei futnetietn5 ref iI17iiEg39t ii rtiet etiee in ptihyeiieiai ptizihleme it else diemenetretee the eiigriiii enre ef the P39D5li il39i tr39EI zems et these futnetiiens For the eeiute at eimplieitiityti wet cen inei DFIM fItt3939thei39 IItquotBa1Erl tECt tit tie the ftitiewing SpE iai ease the mteimhtante ie diispiliaeeti ingte a shape he ffr i39 tIiBfpEit39t dent of the tt t i hiE Bi l1iCii then I EiE 5EIEii frernt rest at the i gi mt t This rneetie that we impose the iriititai eentikitiene e r jD39 fr p and Se i m g i g ten 3 The pm bievm is the de39termine the shape erBt at any euiheeewaierttt time t the U Dttt etreitegfy is the etlept the petttieuer etehttiens already f uind tee the iireii itiiitiei tl ntiiiiii Fitett the part efif 17 that Seyie that the initial ehepe is ii I tdt JEl1Ei39Et ttI ef E irniies that tif is eonetentt so 13 telis 115 that it it the peeititve zettie Vet J r ere derteteci by Al ii3 EL tttett t B t39l tI1El1 iiI retdtteyes the ertey of fltEI39tE Iiit 5 g ED JttHrc tee Amer ci sitii Eetti it 1 12 Next Il8 that if h ettd this iteavee us with cehetiatitt mi ui tipi e5 ef the funetiens rquoti7iL ri we tsinettt ft 1 rb te this peint we have net tt d the feet that smite ef sielturtiene eaft I13 are eiee KE iU ii 15 Aeeetltdingiift39 the ttieet geiierel fetmei eeiuttisene newt eveiiiehiie tee US Hlt39E the in nite eetiee 2 N n39Iij AVnxr P il39 my 19 311 Uitlf fittei step is two tit te satisfy 1 0V ptittirig I o in 19 EltHtIii teqiuetiti the teeuit te fIr M 2 P r H1 The Beesieii E p 5iDi ttheerem Df S li ltt 4 gr39lItiii39i1tquott39I39tfES p this tejpt eSettI tiUm is vieltiidi W39hEnE39h39EI pL is Suitf iettltiy well iJei391air39etdi if the eeef iiettt5 are d e tted lay L M A tfFfi 1 t iiF SOME SPECIAL FUNCTIONS or MATHEMATICAL PHYSICS 377 With these coef cients 19 is a formal solution of 3 that satis es the given boundary condition and initial conditions and this concludes our discussion 4 APPENDIX C ADDITIONAL PROPERTIES OF BESSEL FUNCTIONS In Sections 46 and 47 we had no space for several remarkable properties of Bessel functions that should not go unmentioned so we present them here Unfortunately a full justi cation of our procedures requires several theorems from more advanced parts of analysis but this does not detract from the validity of the results themselves The generating function The Bessel functions 1 x of integral order are linked together by the fact that equot 2 quotquot39 Jx 2 Jxtquot 1quotrquot 1 nl Since Jx 1quotJ x this is often written in the form 5 2lt Z Jnxtn To establish 1 we formajly multiply the two series equot 2 i lit and equotquotquotquot2 E 1kquotk k 3 i0 2 0 kl zkt The result is a socalled double series whose terms are all possible products of a term from the rst series and a term from the second The fact that each of the series 3 is absolutely convergent permits us to conclude that this double series converges to the proper sum regardless of the order of its terms For each xed integer n 2 0 we obtain a term of the double series containing tquot precisely when f n k and when all possible values of k are accounted for the total coefficient of tquot is so 1 xnk 0 cc k x22kn n k2quotquot k 2quot 3939 kn k Jx Similarly a term containing tquotquot n 2 1 arises precisely when k n j 394 Many additional applications of Bessel functions can be found in Lebedev op cit chap 6 See also A Gray and G B Mathews A Treatise on Bessel Functions and Their Applications to Physics Macmillan New York 1952 r3mraEMTInL a Umr 0H5 50 the mm cne icient of F is E DquotT rNar aI 2f T mr 25 N IE 3 wt 1 r and the prmf of 1 cnmp lme 3 simple mnsequemrrse Df 2 is the ddfif n fairmum 2 Q Tu prove this we 4nrrquottic rst haquotlt IEIf laquotl39Erf2 39 Ti EG1frf25Iff I 39n In yIn i pl 39 1233 Hnw cv Eri the prmjugm of the CW3 expmn miais U11 the Il alga I P ZrAr Jayr g G 1 Jur 39 Jlrquot E39En and 4 f ws at ante an equating the iEUE iEiEnt5 of Iquot in the5E enqp1raession5V when r 1 4 can be written as I j W L L C N1 N it F i JJ cIJamp quot in 2 wmI 1 y 1Jkxmym mxaJkm Jampquotl inzxyiniyA 1 2mxJwa QT p E 39 quot If WE r eplace y by E and use the fact that J is even Ur Udd ac mrding as rs is eimn or ddi thclrl JKSII 3fiEalEI S the rVemarka4btle ide VtVitjy 1 Jr 3 f2Jnr4 2 EJg QEE H 6 AwVhi ch ShUW S that IiixN 3 I andl lJr1 E fur H 1 2 E 0 4Be 5 els integral fnnwirmlag when em the mapnnenm an thus laft side emf fbecnmes Ex Sin 9 E4 SOME SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS pKg and 2 itself assumes the form eix sin 0 2 Jnxer39n6 e n CCT Since equot 9 cos x sin 9 isin x sin 9 and emquot cos n6 isin n9 equating real and imaginary parts in 7 yields cos x sin 6 S J x cos n6 8 quotW and sin x sin 6 2 Jx sin n9 9 n m If we now use the relations Jx 1quotJx cos m9 cos n9 and sin 126 sinn6 then 8 and 9 become cos x sin 6 Jx 2 2 J2x cos 2n6 10 nl and sin x sin 9 2 2 J2x sin 2rz me 11 nl As a special case of 10 we note that 9 0 yields the interesting series 1 X 2J2x 2J4x 39 39 Also on putting 6 Jr2 in 10 and 11 we obtain the formulas cosx jx 2J2x 2J4x and Sinx 2J1X 2J3X 2J5x quot quot 39 39 which demonstrate once again the close ties between the Bessel functions and the trigonometric functions The most important application of 8 and 9 is to the proof of Bessel s integral formula 7 1 Jx cos I19 x sin 6 d6 12 0 To establish this we multiply 8 by cos m6 9 by sin m6 and add cos m9 x sin 6 Z Jx cos m n9 n 1 When both sides of this are integrated from 6 0 to 6 II the right side reduces to JrJx and replacing m by 22 yields formula 12 In his P DliFiFEtHEtHT39 L E LEdL t DHt5 sstmn0tmics t Wftfk Bessel E ut tEf d ths fmtcttintns Jls in th farm at these integrals an on this bassist td E EIDpEd ntany at their pr stpertiesquot5 Sums csmitnitsd trastisns If wrists the identity in the farm Jpt39r 8 jpil39x39JI I139lrEt39tI drivitdi tng by J I 39ield5 Jp W 3quot 3P 1 Jp39I Js tfJxi when this tlstmuts is itself applisd to ths second dsttamtitntattnt an the right with p replaced by p 1 and this pmwcetss is tDI 3I 3t7inUt3d sinde nittstly we ut tairt Jpmt p E P J39 x1 1 21 i T I 1 2p t 4 This is an tiitt nitet i tm l l fr Cti il sttpanstiman sf the ratio JJAxfJP x WE E t ffli inssstigats the thtsmfg sf t lt li expsnstinns hers Nsssrtthetletststt may be anf ittnttsmst to point must tihst whtsnt p 0 c H2 h fmltuws fmmt Prtmtblsm pa that JrIgquotIMFJlasgID it t39 IIZ139Lt st Earl 17 This cGnttitnutsd fracttim1s was dit5Ctmrstrisd in 1TM tat Lambert wlm used Ppk tn p1quotCt e39 t 11339t st is itratizrzrnal HE treatsmnetd as t ttll wtsz b Jr is s tmnzsro lquotal iDFt Iquot1 Jt39lquotH i11KBtquot tth n the form sf this mrttitmed fltavrzttittzetnt impli s that rants acsnrtst be rsqtimnaig but tatt rf4 1 so nsiitihet trf74 nm It is ratttimtal Ssttssrtsi mtitnm aws tn tLsmbet t ts argumentt WETEA ptattch dl us by Lasgendts tatatsut 6Y years latsrt 5 FM is uJssttinttrtn Islquot tEsssstIquots strigi1sl tpttrabIsms sass Gtajf and MIst hws uspa A fs H 41 CHAPTER 9 LAPLACE TRANSFORMS 48 INTRODUCTION In recent years there has been a considerable growth of interest in the use of Laplace transforms as anxef cient method for solving certain types of differential and integral equations In addition to such applications Laplace transforms also have a number of close connections with important parts of pure mathematics We shall try to give the reader an adequate idea of some of these matters without dwelling too much on the analytic ne points and computational techniques that would be ap propriate in a more extensive treatment Before entering into the details we offer a few general remarks aimed at placing the ideas of this chapter in their proper context We begin by noting that the operation of differentiation transforms a function f x into another function its derivative f x If the letter D is used to denote differentiation then this transformation can be written Dfx f x 1 Another important transformation of functions is that of integration 1fxl ftdt lt2 0 381 LAPLACE TRANSFORMS 333 Thus the Laplace transformation L acts on any function f x for which this integral exists and produces its Laplace transform L f x F p a function of the parameter p We remind the reader that the improper integral in 6 is defined to be the following limit and exists only when this limit exists 00 b I e f x dx lim equot quotf x dx 7 0 When the limit on the right exists the improper integral on the left is said to converge The following Laplace transforms are quite easy to compute fx 1 Fp mewdx g 8 for x Fpgt dx S 9 fx x Fp faPxquotdx pfi 10 fx e Fp mePe dx P i a 11 fx sin ax Fp equot sin ax dx 2 a 2 12 0 p 0 fx cos ax Fp f equot cos ax dx 0 The integral in 11 converges for p gt a and all the others converge for p gt 0 Students should perform the necessary calculations themselves so that the source of these restrictions on p is perfectly clear see Problem 1 As an illustration we provide the details for 10 in which n is assumed to be a positive integer 13 p2a2 cc 7 px cc one Iix l J equot 39xquot dx x e 2 equot xquot dx 0 P 0 P 0 1 lLxn 1 ii 1 p Lxn 2 n n 1L1 My As this remark suggests we shall consistently use small letters to denote functions of x and the corresponding capital letters to denote the transforms of these functions D1FFEREsHTIM EamT1nH5 It mil be rmmed that Awe have imade E S39E tia39i use TIE1quotE f ma fact that iim 1 mm 5 2 Th above fGrmula5 MIL E6 fmmd in Tabla 0 Sevlt tmn 5039 Additim1al simplg trIansAfmm5 can r3adil39 be det erminEd Vwith ut integratiUn by the i E TEquot 39 at L as in L2x 3 BALM 3Lu 7 P E V II G In llaiter sectMi U ns WE shall de v eLIuup meth ds ftzu VndiVng LapJ1ac Iran31Enrms of mm ummplimateul fumction5 As we sta39tad abwvei the Lap asz tranV5fnrmaaVdimm L C i be rsegardesd a3 tfhc special CREE mf ma gangra integraI tran 5fm matim1 5 1htai ned byr taking 1 P Q G b W and VpJE E Equotquot Why 010 we cmtl se thtsa liIrnit5 and this partivuiar kernel In m39quotd7erV in see why this be a fxruiiitfui chn ice it Vusefugl In cn nsider a suggestive anallng3r with pi391WEf1 se2ries If WE write a power sEri eLs in farm Pl ar1x 1 than Pw natural analog is the IIn 1pnper intcgral J H mix air S L We n w change the nuttat i n slightly by writimg x K and this VinrttLgral b i m J Equot a dsg n which N precisely the Lapl acc transform of the functim u I Laplace transfmrlns are theraf m e the cmtinums anamgs of pmwcr seriues and 5inm pmwcr series are imp rtarxti H anialy3i5 WE Ahave 1reasm1ahEe gmunds far Exp tting that Lapla c39e tVran fDrVms wti al5r ha in1 p1tant C sham accvm4unt of LAapIacwe is give1n in Appmdix C Evail1ate th iniegraE5 in V 139 flglj 12 and H3 p Withuut iI1 3gIEItiI Jg Vshmw that H 1 1 La L5E11h I 2 3 F H h L cush M V 1 iI I P2 E1 J I LAPLACE TRANSFORMS 385 3 Find Lsin2 ax and Lcos2 ax without integrating How are these two transforms related to one another 4 Use the formulas given in the text to nd the transform of each of the following functions a 10 d 4sinx cosx 2equotquot b i x5 cos 2x e x sinz 3x xquot cosz 3x 2e3quot sin 5 5i Find a function f x whose transform is 0 A30 1 a 39 d p4 p2 p 395 2 1 b 4 P 3 p4p2 4 6 C17p24 6 Give a reasonable de nition of 3 49 A FEW REMARKS ON THE THEORY Before proceeding to the applications it is desirable to consider more carefully the circumstances under which a function has a Laplace transform A detailed and rigorous treatment of this problem would require familiarity with the general theory of improper integrals which we do not assume On the other hand it is customary to give a brief introduction to this subject in elementary calculus and a grasp of the following simple statements will suffice for our purposes First the integral 3 xfxdx 1 is said to converge if the limit b lim f x dx b 0 exists and in this case the value of 1 is by de nition the value of this limit Lxfx dx J0bfx dx Next 1 converges whenever the integral Ox lfx dx convergg and in this case 1 is said to converge absolutely And nally 1 converges absolutely and therefore converges if there exists a iFFEREHTLeIa EEU Teii H S fueneeetieeen g xj such that f1 gx and f my zn eerwerges this is known ea the eeempereiee reeiijie AeeeIdinIy if fxf is a given fur1et39ien de ned fer 1 E J the C 1VET El1rCE ef 1 reqeuiree first ef am that the im legrel S 15 it must EJ39iEI fer eeeh enite gt Te guereeenteee this it sieujf eee to assume that Jar is eentiermeeues er at least ie jfJixE vEWE E CDJ iHaF39EE v H Te By the letter we lneen theet 35 ie l3Uil339lEi llquotUS ruwer ewerF nite intewal G E 1 N b eexeeeept peeeisbly at e rslite numbieer ef pe ints wihere there are jump edieeeneetineuitiees at ewehieh the fUI1El39i FI app1reeehes dgi erente Emilie fmeme thee left and right FigMree 53 i1luetreatee the eppeelreneee ef e typieeel piecewise centi1nu eue funetieng its imegrel frem III in be is the slime the quotinitegrelse ef its eeentinueee parts U Ir El lquotquot the eer1reependin subieneterveelee This class of funetiem IE mIm39 i139m39E eiIquottuelNy aw tehet ere leikeiy to erzieer in preeeteiee In p f iEiLII I it ineluedes the dei5eentineemI5 steep feeetieens and sawteeeeth fll tisUHS eepresssieneg the eudwiien eppleieatiene er remewe ef feemee ened Veltees in pfeblems of pehysies and engimeeeereinge H ee is pieeewise eentinueues quotfer 1 2 p the enly 139emeir1ing threat te the exieteen ee of its Laplace trenesferme F p 2 ft e39F fr ex is the behevier ef 9 eimerend equot fr fer llerge x In Order te make sure that this integrand dimiinishes rapidly eneeegeh fer eee n veergeeneee eeer that fx ees met grew tee repM1y we shei further eae5eme e etheft f is j 1 x 5 LAPLACE TRANSFORMS 387 of exponential order This means that there exist constants M and c such that If IN 5 M6 2 Thus although f x may become in nitely large as x gt 00 it must grow less rapidly than a multiple of some exponential function equotquot It is clear that any bounded function is of exponential order with c 0 As further examples we mention equot with c 2 a and xquot with c any positive number On the other hand equot is not of exponential order If f x satis es 2 then we have quot fX 5 Me quot and since the integral of the function on the right converges for p gt c the Laplace transform of f x converges absolutely for p gt c In addition we note that s equot fxl dx Fv f equot fxdx 0 M S M equot 39Cquot dx p gt c 0 P C S0 Fp gt 0 as p 3 Actually it can be shown that 3 is true whenever F p exists regardless of whether or not f x is piecewise continuous and of exponential order Thus if p is a function of p with the property that its limit as p gt 00 does not exist or is not equal to zero then it cannot be the Laplace transform of any f x In particular polynomials in p sin p cos p e and log p cannotbe Laplace transforms On the other hand a rational function is a Laplace transform if the degree of the numerator is less than that of the denominator The above remarks show that any piecewise continuous function of exponential order has a Laplace transform so these conditions are suf cient for the existence of L f x However they are not necessary as the example fx xquotquot2 shows This function has an in nite discontinuity at x 0 so it is not piecewise continuous but nevertheless its integral from 0 to b exists and since it is bounded for large x its Laplace transform exists Indeed for p gt 0 we have Lx l2 zf e pxx12 dx 0 and the change of variable px t gives Lx I2 Z p 12 e ttl2 dz 0 LAPLACE TRANSFORMS 389 W I I I I I I I I l I We I I I I I l l I I I F 1 FIGURE 59 we have x dx 1 Show that 1 e LlfeXI 39 p6 and p7 1fex 1 Strictly speaking limfx does not exist as a function so Llimfx is not E 0 e 0 de ned but if we throw caution to the winds then t5x lim x 4 is seen to be some kind of quasifunction that is Iin nite at x 0 and zero for x gt O and has the properties 00 6x dx 1 and Lc5x 1 This quasifunction c3x is called the Dirac delta function or unit impulse function2 2PAM Dirac 19021984 was an English theoretical physicist who won the Nobel Prize at the age of thirtyone for his work in quantum theory There are several ways of making good mathematical sense out of his delta function See for example I Halperin Introduction to the Theory of Distributions University of Toronto Press Toronto 1952 or A Erd lyi Operational Calculus and Generalized Functions Holt New York 1962 Dirac s own discussion of his function is interesting and easy to read see pp 5861 of his treatise The Principles of Quantum Mechanics Oxford University Press 4th ed 1958 A390 h 1FFEaEH39TIm EDUAT NE 5 T0 Shuppijse we wish tn nd the parthiacular smuthian njf th hc ifferanhtialh equa nn M W w by v 1 q satis es the inihtial wnhditians M0 y and y39U y It is cflheahr that we cuuld My tn apply the methndhs of Chapter 02n tn t nhd the general snluhtimnh an thehn evaluate the aarhi trary crrmsta nJts in acmrdancampA with the givan inixtial Ccwndiliuns Ii wEver39h the use Uf Lhap hahce transfazrnngs pmvidcs an aJternaEe way anrf attacbiihnhg this pmbl tn thath has several advanIahgE5 T0 SEE haw this methnd wmxksh let us apply the Lap acie thrhahnhsfohrma Iiz rl b hthcjl bath sidcs 0f 1 my M1 LfI By the Iineahrhity if E this can has written as Ll9quot HLhy b lyl LfZhrI 2 Dim next swap is to express Y and in mrhms hnhf hFihrst an intheghratimnh by parts gihvhs Ill L M Equot yquot dx quotI ye 39 39 pf EquotP39 y dx Y 1 y phL fM My 3 Lh h h i E Lh y Pi Wi PlW LW P1L pWhh hh 4 we rmw insErt the gi139 a39eI1 initial mndili zm5 in and 4 andh Suhstituthe thheset eiuzhprassimxls in 2 we Thtaihnh iarn algahbhrhaic Eqjuatinn fohhr y PEL M 22 wilyhll mm h Lv fx7hh and shmihving for yiesls LU E Lfxg2ip iH 4 3 5 so N e M S0 The fuarmtionh x is known 5 its LaplaaE tramsfmrm x is 3 SpaE r f fl fi fll nf p and S im E ha b m and v are knmwn crgmshtanis Ly is conmpiavlze y knuwh as H hfhuanchhthihon af pr If we can naw nd which fumtinn LAPLACE TRANSFORMS 391 yx has the right side of equation 5 as its Laplace transform then this function will be the solution of our problem initial conditions and all These procedures are particularly suited to solving equations of the form 1 in which the functionquot fx is discontinuous for in this case the methods of Chapter 3 may be dif cult to apply There is an obvious flaw in this discussion in order for 2 to have any meaning the functions f x y y and y must have Laplace transforms This difficulty can be remedied by simply assuming that f x is piecewise continuous and of exponential order Once this assumption is made then it can be shown we omit the proof that y y and yquot necessarily have the same properties so they also have Laplace trans forms Another dif culty is that in obtaining 3 and 4 we took it for granted that lim yequot O and lim y e quotquot 0 However since y and y are automatically of exponential order these statements are valid for all sufficiently large values of p Example 1 Find the solution of W4yM that satis es the initial conditions y0 1 and y O 5 When L is applied to both sides of 6 we get Llyquot 4Ly 41x 7 If we recall that Lx 1pf and use 4 and the initial conditions then 7 becomes 4 f P54HH s p 0139 I 4 U M p5 SO p 5 4 L D V4r24fm P 5 1 1 p24p24p2 p24 P 4 1 2 2 7 p4 p4 p 8 On referring to the transforms obtained in Section 48 we see that 8 can be written Ly Lcos 2x L2 sin 2x Lx Lcos2x 2sin2x x 0q n1ee39eeieirerm t EQlLJATtt1H39E se 32 E3C39S2ifi Zsiinlx 1 is the desired E 1tttiiDn We can easily eheek this sesiufiti for me tget1ersi iseltttirn mi 64 is SELEIl39t tang i ZJ EETiDr te be y e eeslr ea 2 x and the initiel eenditiens simply st mice that isquot 1 end e e The sreltiditty estquot this preeedute eleeriliy rests en the sseumpitieani ltheit only me funetien ix hes the side eat erquetiten 34 O its LeIece iiIquotE1 f 39iTl tThis is true p we restrieit euitseiees te eeminuiaus yix s s sndi en seiliutioni of e cli ifereintiatflt equetie1n is necessarily eeintiinueus Whien fx ie assumed te be eenttintteue the equetien lx Mp is ieftien written in the m T FTFU ifHl It is eusterniety te eel L39 the immerses Lepteee F i af rH1i iE i V and to refer tie fr es the iH39UE f539 E Lepiieee triemfertmr efi e Since L is linear it is esiemt that L4 else liineseir in Esiempie 1 we tnede use ef the feilewii1g irwetse Il39 Sf Clm5i 39 H I 39 t Lquot39 sap eels o L 1 sin P L391 E 2 pa p 5 p This example else iiiusitt39sit es the value efi deeempesitiien into pettisi fiEa1Eti39U nlS as at mettied tilf nding inverse trsnsterms Fer the ieetwenientkeie ef the teeder we gsise es simrt list ef iusiefui tre1nsferm pairs in Tsbie 1 mete IE3tIE39E1SiquotuquotB tsbles site sssilable fer ti39tE1 use ef theses whe nd it desitehie tie stpipily iLep39Isee tI39 t Sf i39lT15v fteuentily in theit sweetie We shell eensidet e tl3939l1F b 17 ef general prepertieis emf Laplace transforms tieistty i39Il39Ct39E SE the ieiisiibiliitty q Table 1 The titer est ithese is the i hEflli ig fermiuie iiE fLrHiFTes i 91 Te xestebliish ithis it su ees tie ebsewe that I31 ewenfewewee E 3 e t i fx Li H Fw l Feitmulis AE9 sen be useu1 te tied trsnsfermsi ef pmduets ef the ferm iemf x whien Flip is keewin and also te tine i1werse E r39 S7iiUmS ei tueietiievns at the term e iwi1en fr is itnewn DIFFERENTIAL I Uamp39JDNS PR39lBLEMS 1 Isa Afa i I L Fii ii tl39m La p IacE tiranvsfmmg Uf X Ve L Cb I211 IE c 63 ms 5 inxrerse La plmVe tzransfncrrns if D j gtp 21 9 M P 33 G p 3 5 ti Ep P g H y393939 y 313 Jr P SmLl39vE each of km f liluwing d i Erenrtial aquVatmampns by the mmhnnd of Laplace tran5fUrrnjs p y 3 y 3 3quot Lb J 39quot 41 4v EL quotU 398 and M3 3 W 0 U h mar 0 U P wJ 1 E y quot y Ir iv 339e quot sin 1 yf li and y p C the solution crf y Zaryquot may 2 E in which the initKial cmidiitiIzuns yI39 yg and y D ya are left unruEsatrictedV This pmw3Av39i4es u ad1ditinnal desrivatinn nf DUI ariie r 3uIutVin w1V in Sectinn 1 for the Acrase in w hicih the au4xi1i5arquot uquaatiun M5 3 dmuhlte mm M0quot D VAp pJy 3 In as Iah ii5h Elm fmmiula for Eh Lapa cT Eransfnltrm of an i ntEgra1 1 rm map In J J and uEr if3J this by EINITIliifngf 1 N M E im39l t39wn u way5 P SEILE r39E y 43 f ydI39 6quot quotU E 0 Cm1sidteltr the g f i Lapae transform fmrn1m1a V 1 Fm W f 2 E f 94 The differVenIsiatVi n Bf this with respect to p dtf the quotlIquotI ErEg gIquoti39L5l sign can be juSEi a6C3L arm yield Em I H our m LAPLACE TRANSFORMS 395 or Llxfxl F39p 2 By differentiating 1 we nd that Llx2fxl F quotPL 3 and more generally that L1quotxquotfxl Mp 4 for any positive integer rt These formulas can be used to nd transforms of functions of the form xquotf x when F p is known Example 1 Since Lsin ax ap2 a2 we have d a Zap quot 39quot quot quot2 F397239 t Example 2 We know from Section 49 that Lx 2 VJp so U LIX2 Lxx I2 i pP If we apply 2 to a function yx and its derivatives and remember formulas 503 and 504 then we get d 43 Llxyl quotELIH quot dp 5 Lxy l 5 jf Lty391 gtpr yltogt1 tpY1 lt6 and d d Lxyquot1 Lyquot1 3 W1 mo y39ltogt1 s f5p Y pyron lt7 These formulas can sometimes be used to solve linear differential equations whose coefficients are rst degree polynomials in the inde pendent variable Example 3 Bessel s equation of order zero is xyquot y xy 0 8 It is known to have a single solution yx with the property that y0 1 To nd this solution we apply L to 8 and use 5 and 7 which gives d 41 quotquotquotlP2YquotPlPYquot1quot 0 dp dp or dY P2 1d PY 9 P D39FFEEE hFF L EUIaTED H5 If we skerparme the variables in 9 and in t gmt wt gm 10 Din expanding the last tacIutr by the binomial serias I 2 E 1 HE Bag 52 mm 5 a f 2 In 4 pn 1 W J H 13 gt7l I bemmes 1 an E 39l I 139ff J Tn Apg H A 33 i Zn 21 If we rmw gmr ecd fnmaHyi and canmfpma the i quotE I39EE 39tr39arsftJrm of Ehis ErEI ii 5 term by I fmi than we nd that v In jg 391 f 39S 22 23441 214 53 A Si ti M3 1 it fTDITE1w5 t hat f a 1 and mr s il39utimn is 39 39P 1 1395 gt 0 p o pk K This SEriE5 d nes thE importmxt Bessel function J x4 wlmse La pEace ilimansf uzrrznm we haw fuund to be If pf We mfbIainef this Begrines in Chapter 31 in a mta1y di erent way and 0 is Aintemzstirng Vtn SEE hcnw easily it Can be EIer39Eved by Laplace tran5f Drm meth ds W rmw tum to the pmbilem cavf integnatiing transfurm sz arid wmr main r sult is L7 J Ftpldpe MA 39 P LAPLACE TRANSFORMS 397 To establish this we put L f xx G p An application of 2 yields 4ew 4mvn nm SO P mm fnm for some a Since we want to make Gp gt 0 as p gt 00 we put a co and get 01 rFpdp which is 11 This formula is useful in nding transforms of functions of the form f xx when F p is known Furthermore if we write 11 as we Kitzdx fcFpdp 0 X and let p gt O we obtain Lxgt dx f0wFp dp 12 which is valid whenever the integral on the left exists This formula can sometimes be used to evaluate integrals that are dif cult to handle by other methods Example 4 Since Lsin x 1p2 1 12 gives sinx dp Jr f dxf 2 tan p 0 X 0 P m 0 2 For easy reference we list the main general properties of Laplace transforms iii Table 2 It will be noted that the last item in the list is new We shall dis uss this formula and its applications in the next section PROBLEMS 1 Show that P2 a2 Lx COS ax gray and use this result to nd L llt73 2 2l P MFFEREHTML Emummws TAE L E Genheral pmpertiies at LEULEJI PIP Liilcr xl gixilla FP Whipl L E fhxi FLU H Lhhf EhW PFEIPJ fitquot L LWEIAE E piFftp pfU J H03 If mM l MmH FTM MMWhHFWm L E J FIpdh I I illll nxwmmHmmm 1 Find Em h h f H f 391ailwhi39ng rriansfUrm5 3 L1f339 sin T b 1 S hre Eagtish f the fnlicawing hdhi quoterehr1Iia Eql1a i I3 Iquot 31 TN 44 9 E 393 JW W b Jyquotquot quot I 3 39quot yij U 4 If yIc satis es the di ehrenIiaN equali n p Wh f MAUI Jh Md quotU E 95 zshmw that in transf rm Yip 753 Ii5 33 the hquhaIiunu Yquot pg py ygh Db75E1 VE Khan the sec nd equa391ian is 1if the same typ as the zrsti sh that no plrtgress has been made The meth d of Ehampflt p is arvam1htag u5 uni wh n the me icrihenhts are rs damp6quotgrEE hp39 ll3rIrmmials 4 If cs ampI li 0 are pttsritiwa c nmh5htant5 evaluate the fhEhJw39hihng ihmgra15 rm A at MI 3 J TE h E 0 n 33 E E E 15in bx M U1 e r KShh0w hf rmalily timer 39 F a Tux n FE E P cm x 05 0 fat 131 LAPLACE TRANSFORMS 399 7 If x gt 0 show formally that ltagtrltxgt quotdz 5 0 t 2 g cosxt 5 x b fx 01t2dt 2e p a If f x is periodic with period a so that f x a f x show that Fp 1 1eap 0 e P fxdx b Find Fp iffx 1 in the intervals from O to 1 2 to 3 4 to 5 etc and f x 0 in the remaining intervals 52 CONVOLUTIONS AND ABEL S MECHANICAL PROBLEM If Lfx Fp and Lgx Gp what is the inverse transform of F pG P To answer this question formally we use dummy variables s and t in the integrals de ning the transforms and write ac FltpgtGltpgt reWits as U eP gzgt at 0 Lee ce fsgt ds dt J U e p fs dsgt dt where the integration is extended over the first quadrant s 2 O t 2 0 in the sitplane We now introduce a new variable x in the inner integral of the last expression by putting s t x so that s x t and it being fixed during this integration ds dx This enables us to write FpGpl Uie quotfx tdxgtdt J fequot fx tgt dx dt This integration is extended over the rst half of the first quadrant x t 2 0 in the xtplane and reversing the order as suggested in p l DIFFEEE1E TML aqua rmws n 39 a l pFi 60 we get i p E amt aw dx in 2 Km gFi Hjf x E3 If nit 3n 2 L gm 1 Th Vintegrail in the last EEpr39EEiUn p k a funxcticrn of the up r lirnit x and p mvidBs the answier tn n 39ru1 questi DrL This integml is raMed tha 39 F1 Un quotJ39Wf of ME funicti 0Vn3 fr and g It can be regard ed as a gemeralizEd prtdu ct 39 mf thesE fu mcti0nsV 0 c fazst stated in eqVuati n 1na 1Eiy that the prfoduct Hf thaa Laplace transfrVms if tw f unmmVns p thg Vtransfm5m of their c nVmVlutinn i5 callgted tha wnvmLm39a3n heuwmi The nn39v1uJtiUn th em39em can be usa3d U i nv ers trans39fr1rm4 fFm39 LinsmnscsE 6 x 2 Mpg and Lsin Jr 1fp3 W O we have Llp2pE1 DJ E LEl I m 3 sin If at Q I sin r as can easily be veri ed by partLiaI framinus A n1w1re intEresiLing class uf applicmins arises as f UWS If fJr and kx arae givan fuim1i0ns than LAPLACE TRANSFORMS 401 the equation fx we fax rgtyltrgtdr 2 in which the unknown function yx appears under the integral sign is called an integral equation Because of its special form in which the integral is the convolution of the two functions kqx and yx this equation lends itself to solution by means of Laplace transforms In fact if we apply L to both sides of equation 2 we get Lfxl Lyxl LkxLyx Lyxl The right side of 3 is presumably known as a function of p and if this function is a recognizable transform then we have our solution yx SO 3 Example 1 The integral equation yx x3 J sin x tyr all 4 l is of this type and by applying L we get Llyxl Lx l LlSiI1xlLlyxl Solving for Lyx yields L 3 31 4 Lyx1IECsinx1 1p2 1 P 3 P21 2 p4 P2 P4 P6 SO 1 yx x3 5615 is the solution of 4 As a further illustration of this technique we analyze a classical problem in mechanics that leads to an integral equation of the above type Consider a wire bent into a smooth curve Fig 61 and let a bead of mass m start from rest and slide without friction down the wire to the origin under the action of its own weight Suppose that x y is the starting point and u v is any intermediate point If the shape of the wire is speci ed by a given function y yx then the total time of descent will be a de nite function Ty of the initial height y Abel s mechanical problem is the converse specify the function T y in advance and then nd the shape of the wire that yields this Ty as the total time of descent 4013 mFFEHEr rriAaL EQLIATIDWS F 1 quot39 i a Q 31quot FI G39URE 61 Ta furmulate this pwbhem mVathmatic31l we 5tar39t with title pri39nsipf7le Vmf cnnservati04n Uf en rgjy 1 d3 M2 E E W 1 Err E T q which can be written as On integmtifng this frnm U s y m U G we get 3u Uzy Hy J dr I P v y 39ua n Nnw I n n H R ES alga krmwn If wg in5er39t P in 5 than WE SEE Izhat 1 my 0N P S and this enahIes us DR cralru1atE Ty wVhen ever the curve is given In AVbel 5 pmbl m we mam In End the cum when Tiff is given ayrld mm LAPLACE TRANSFORMS 403 this point of view the function f y in equation 7 is the unknown and 7 itself is called Abel s integral equation Note that the integral in 7 is the convolution of the functions y quotquot2 and f y so on applying the Laplace transformation L we get 1 LlTyl LlyquotquotquotlLfyl 2 If we now recall that Lyquot 2 Jrp then this yields Llfyl v If 7 I 1 35 12 p LTyl 8 When T y is given the right side of equation 8 is known as a function of p so hopefully we can nd f y by taking the inverse transform Once fy is known the curve itself can be found by solving the differential equation 6 As a concrete example we now specialize our discussion to the case in which T y is a constant T This assumption means that the time of descent is to be independent of the starting point The curve de ned by this property is called the tautochrone so our problem is that of nding the tautochrone In this case 8 becomes 2 2 T Llfyl pquot2Ll7il Pquot2Equot bquot2gr where b 2gT2IE2 The inverse transform of Jrp is yquot 2 so fy 5 9 y With this fy 6 now yields dxz b lt gte dy y as the differential equation of the curve so b xf ydy Y On substituting y b sinz cf this becomes x Zbfcoszdadqb b1 cos2c1gtdc gt g2 gt sin Zqb C l1GURLE 62 30 22rb em 2m at end 3 t ELI ees emg W The curve meme peee threeugh the erigein 0 Se C e P and wV we put a M2 and 3 2 then I0 take the eimp1Ler ferm 1 am sin J and LT afl ens 8 These are the areemetreie equatiene ef the eyele id ehewn in J P M whieeh is generated 3 e e xeed peint en e circle emf radius e mlleing under the heeJrizeent aLeII deesheed lliine y 2a Since 2e b Eli the diem eter ef the genere1ieg eirefle is elietermineedi the emleftent time erf dE5c ti AeeD3r dingl3r the Icea ut eehrene is a cey ce1eiede In Perebleeme A and 1115 we aserii ed this prepe rtey ef eye eiede by ether mzetehedlez Our present di5eu5eiJ een has the eeidveneetasgee ef enahling US In ned the teeutee ehr nee withtmt knewim p 7 edvaneee what the answer willi be 0 5 9 Find L 39Ifp39I39 elf 13 eenv eiutieen See Pmblem yt 6 Salve each ef the fe eeing iemetegral equeatiene 1 ywe 1 3 ax e ms em 7 h yr e 1 J equot I dr 1 n U3 e me TZJI we I H et y39 dr e d 35in y x Jffx e ym dt LAPLACE TRANSFORMS 405 3 Deduce T gtd 0V Jr dyo from equation 8 and use this to verify 9 when T y is a constant 72 4 Find the equation of the curve of descent if T y ky for some constant k S Show that the differential equation y azy fx y0 y 0 0 has yx 31ft sin ax t dt as its solution 53 MORE ABOUT CONVOLUTIONS THE UNIT STEP AND IMPULSE FUNCTIONS In the preceding section we found that the product of the Laplace transforms of two functions is the transform of a certain combination of these functions called their convolution If we use the time t as the independent variable and if the two functions are f t and gt then this convolution theorem equation 521 can be expressed as follows LftlLlgtl L fltz rgltcdr 1 It is customary to denote the convolution of f t and gt by f t gt so that rrgtltgzgt fro ogtodr 2 The convolution theorem 1 can then be written in the form Llft 8tl LlftLlgtl 3 Our purpose in this section is to discuss an application of this theorem that makes it possible to determine the response of a mechanical or electrical system to a general stimulus if its response to the unit step function is known These ideas have important uses in electrical engineering and other areas of applied science Any physical system capable of responding to a stimulus can be thought of as a device that transforms an input function the stimulus into an output function the response If we assume that all initial conditions are zero at the moment t 0 when the input f t begins to act then by setting up the differential equation that describes the system operating on this equation with the Laplace transformation L and solving for the transform of the output yt we obtain an equation of the g DFFE3REN39l uL EuLmT39nNs fnmi 0 0 gm 14 when z p a p U1yn0mial whrse r0E ici mts dEpBnd vsmlly an th 539 paramet ers of 0I system itself equation is main snurca of the expIMicit fm m39ul4a far y39r that we 0htainT below with the aid of th c0nv0 1 utim1 them5m Let us be mm speci c We seek 5nEutinn s y DIE the Vlisnmr di1 ere111ial equatinn P m m tlmt satisfy the iniitial nnd3itimus PiU v 39ZUJ U 6 describiVng a mechanicall or electr ica sVyst4em at rest in its equilibm1m piw5iticn The iLnpm f can be t1m ught of Oc an imipr s d eAxtetrnaVl tVr e F D1quot e1ztr 0 m04ti4ve frc that b gin ta act a1timEI39 Y as disvussad in S7EeE ti F a WhEn this input is the unit step fur1cti0m Aum de mmd in Pmblem 49a2a the 5trEu39tim1 r output yr is dannted by g and called the f1dfCf f respanane that is a M W By ap gpl ying the Lapllauce transfm4matit3n L and usVir1g 39f f L1iA S 3 and 4 in Swtion SH we obtain PA2U 4 F D 4 Ab LA r4 L m mfi E E 50 W1 1 1 an 1 W Pt T Q b 3 Ezm whare Ep i5 de ned by the last eq uality We rmw apply L in tha gamma way ta th6 giEnmaI eqjuatrigm 5 whmh iEI IZlS 4 and dfhiing bmh sigdes Bf this M p amd usigng W giveis 1 E guy P Lm fF H ltlfle lt8 The mum1u tiDn thenrem n w Ena bWe5 us to write 8 in the farm guy LAVwrrAJ L Ac rJr mltdr 1 LAPLACE TRANSFORMS 407 By using formula 503 once more we get Ln pL039Altr rfrdr L At tfr dr so d t yo d fan tfrdr 9 By applying Leibniz s rule for differentiating integrals to 9 we now have ya 39A39r om dr Aroma 10 Next since LALf LfLA 8 also enables us to write 3Ly LlftAtl LU39fr agtAltagtdo P 0 and by following the same reasoning as before we obtain yo f t owe do f0At T 11 0 In formula 10 we notice that A0 0 because of the initial conditions 6 and 11 takes a more convenient form under the change of variable t t 0 Our two formulas 10 and 11 for yt therefore become ya fA39ltr rfrdr lt12 and l I J1t Tf39TldI f0At 13 Each of these formulas provides a solution of 5 for a general input f t 4 Leibniz s rule states that if Ft ffj Gt x dx where u and v are functions of and x is a dummy variable then quot d d Ft L Gtxdx Gt vd J Gt u1 lS See p 613 of George F Simmons Calculus With Analytic Geometry McGrawHill New York 1985 mFFssfsHTisL E ampLIsTiiis in terms of the indisial rsspcmiss AI in the iiniiz srtsp iunstimni FCIil mUia 9 is scimeiimes csHsd the prii nicsipis sf 3iHp39KEfp 39ff it has iJquotvEE139i vsiiiausiiy sttiriibiitisidi ID the f mCHLIS nsiinieieeiith sntur3 39 James Cisrk Maxwell and Ludw39iig Baiitssmsrin and 3157 in hrs English appiiisd l Hi1EI iIiCiEH Ofliiisi Hr V39i5idE Illli Example 1 Use fmmuisa 1 3 tin Stliiif r 6y 2 where ym FWD T U iiquotIEiE we i quot3quot quotE 1 R R 513 by psniisi frsctiimis and iiiquot39 39EI39L5 iDn we nd that 3aims EEs 3 E s39 since jm P m sis arm fin P 13 gives J 1 s 1 ii I 1 211111 1 em i T quot EI 39t 39quot y rj J E we me A s 1 2gg Z iE1r s 15 IU K 39EI1r E as Eir 39 15 5 susiisiitsiion vsri be ssr39i sd by subsiti1i ut ing dirzscstiiy in tins giiir si1 imlqllE3itiimirliquotI srnid EIISEll 0 sisrisinig this Esqli ti i by this mmiin if z iji sliidiisd in Sssttiimi We scan also Lise mrnziuils tuai Sa iilif the isquiatiinn in this sssmpie but bepfsis dissing A it is desirsbis to Esp39rEssi 123 0 W s SiirifL39Ii6I iarm W srciDmipiisih this LiSi g iiE3 39llI1i39i impilise fumtzti n described in Pmiziiieisii 4925 In phiyisiicis this impuiss due in s czizrnstsnt force F acitimg user s time i niervsl at is ds nsgdi ta pa M The 39fUl C39iiD W can as 39tiiicziiighii Eff sis s limit iiif mnsissint f iJI7IiCiIiU39fliS of uriii ii39TlP39Jii5E acting nvsjr Si39IDIquot39EI and sh t1rter iiniiswisis f isimiEi it is used ti des cribe fnmss and V il g i M sci irezry ssii ddeniy as in the E i af 3 i39li rlTlliquotiquotiEiquot binws n is misciisinissii system EFT s iigihiniing S iIquotDkE on a tI Ii1IL5Tii55iU line Fm issi 1114 Essential puTUP39Elt1quoti39 ijjf V is that szriipissssd by ihs squs mn Li twi 1 bi iffi di in Pmbiism A iwheii the input I in this ii emntisi sqquotustimi 5 is she 39I1riii imipuise fLinCiiia n tins wigl39EPgtl 39 Mr is dsisnnte LAPLACE TRANSFORMS 409 by ht and called the impulsive response Applying L in this case yields 1 Lht Z5 14 so 1 1 ht L Am By 7 and 14 and it follows from Problem 505 that At J 1tdt This shows that A t ht so formula 12 becomes yt fht 39rfr dr 15 Thus the solution of 5 with a general input f t can be written as the convolution of the impulsive response ht with f t Example 2 Consider again the equation y y 6y 2e3 solved in Example 1 We have ht Lquot 1 1em e 3 P 3p quot 2 5 so that 391 2 t39 quot5I t 3r yt Se e 2e39 dr 0 1 1 2 563 Be 3r 3621 as before Remark 1 In complicated practical situations electrical engineers are sometimes compelled to work with indicial or impulsive responses At or ht that are only accessible experimentally by means of oscilloscope pictures responding to generator produced step functions or impulse functions In such a case the output must be calculated from 13 or 15 by methods of graphical integration that permit the plotting of individual points on the output curve For a discussion of these topics see Chapter 9 of W D Day Introduction to Laplace Transforms for Radio and Electronic Engineers Interscience New York 1960 419 DJFFERENTlAL eeLmTteHs Remark 8 Te fern e mete generefit View at the meaning Elf ee nvehttien let hue eeneidet eh linear physieel eystettmt in twhttzh the e eet at the pteeenyt 39time I hf H smelt stimulus gt Ci at any past time 1 1 prepterttie1nalt tn the size ef the stimuthzst We fLtrthet39 ampiSStl 39le that the prepertienahtquotfy teeter d Ev39EI tdSt ehly en the telepeetd time t I and thee has the term fte t The e feeet at the pl39BSEt tt time tee thereteare f EJSCAFJ d Sginete the systetm his ltineer the total e eet at the preztsent ttitee t due to the stigmuhte acting thFDltghDH39I the ehtire pest l397tiSt t f5r ef the eyetetmt tie eibteianed by eddihg these S pi l itti e eete aridquot this leatts te the tZln hquot l tIIii T inttegtel Em The tewetr tirmt here is G heeeuee we essutme that the stimulus etart ed aettitntg at time t 0 that is that gr fer 1 1 1 The imptertentee ef etentwzztut ietnt tie di iEtLllt te etxeggetatte it premittetst the reeeehehhte way emif tekiirtg eeteount of the past in the study ef wave metxien heat eettdT39uetien dei hstietn and ether eteee mtethemetieat phyetes I L Sheet that ft 1 g t gm t ditreetlgt fretn the det hitien 2 by in39ttredIte irtg e new dert1tt1jy39 vet39iehlEe er t t This ehhsewe thete the Dp fF Lt39iDtnI Def fetrmriing eerwelutietns is e etmmtutethre It ie atee aeee etiatiee and dietributive rm hm W3 mm teat W1 etttzi tr hm WJAAI gtfttt eff 4 Up htth fw Wt WJA PZ NJ hilt 4 etlt ti Mttlet n rintereetirtgt diseeeei en ef the abstreet I JT PE1l39Ii ESx ef veenvetutten P given by Mark Kate eret Stenietew mam en ll4 U 1d2 ef Methemettt13 P teed LgEe New tmerieen Lttfhtreryy New YeI k 1969 if Find the emreluttietn ef eaeh ef the feHew39ithg pairS ef funetetetne ire 1 einet J e teaquot wh ere he hi I5 Emi 0 5hrtet sin ht wtthtete e M ht iquotu1 f 39tfquot39Ijt the t DIT39v39EJIILt Eit In etheerem fer eeerh ef the pairs ef furtetien eeneid er ed in 0 0 4 l Use the methedfs ef teeth Ett mpiE5 I end 2 the seh e each ef the feHew39ing MT di erearntietl equetientst 0 yquot 5y z t5e y 39 2 ye Gt 01 v pc t 2 fr M3 J i i 0 39aquotlt Jht y H y g cm M0 2 U LAPLACE TRANSFORMS 411 V5 When the polynomial z p has distinct real zeros a and b so that 1 1 A B zltpgtquotltp agtltp bgt pa pb for suitable constants A and B then Mr Ae Be 39 and 15 takes the form yt JfrAe quotquot Be quotquot dr 0 This sometimes called the Heaviside expansion theorem a Use this theorem to write the solution of yquot 3y 2y f t yO y 0 0 b Give an explicit evaluation of the solution in a for the cases ft e3 and f t t c Find the solutions in b by using the superposition principle 13 6 Formula 13 can also be derived from 4 as follows without the use of Leibniz s rule for differentiating integrals Lmrgt1 1 75 39 5 LlAtl pLmrgt1 LlAtl Llf tl Mo LlAtf Il rltogtuAltzgt1 LUlt rf r dr f011 Llytl PLlf 11 Check the steps 7 As we know from Section 20 the forced vibrations of an undamped spring mass system are described by the differential equation Mxquot kx ft where xt is the displacement and ft is the impressed external force or forcing function If xO x 0 0 nd the functions At and ht and write down the solution xt for any ft 8 The current 1ti in an electric circuit with inductance L and resistance R is i given by equation 4 in Section 13 Li R Et dt where Et is the impressed electromotive force If 0 0 use the methods of this section to nd It in each of the following cases Et Eut lttspgtiEltzgt 12 am c Et E sin wt a1 tilisEelteHe39tlsL eeestieits p8 p8 k Pierre Seimen de Laplace l 74 9 Al82e7 was a F39139El lEl e mathematieeeieeaene and theetetieal astterlettiet whee was she fattlells in his ewe time that he wees l l39lDW39I39l as tlie l39lewiteiee ef Ftianeee main l t reste thteugehequotut his life weree eelesltial l11tEl lEt lC5 the theery el pteihaelhielityl and I2Elquot39e5IEIlquotla a 1Zil39ll3l lE iEl39l7leE17l ll At the age et letwl lllijfef f he was eler dj39 deeply etigeageed in the detailed applieeatlee ef Newterfse law ef grasita tiiee Ie the EDlal39 system as a wheilee itl etlhieeh the platlets and theirs satellites are met gmret l1ed by the sun elerle hut ell lliEIquot CI39 wlrth ene at1e39ther in ea l3EW lldE l g sarlteety ef ways 7Evet1 Netwtteinl had been f the epinlen that eivitie eliitetsentieitnl weulldi DE393 Sl Di39l llejl he te eded te pteseneti this eempleit meehanllsete fretn dei gIE39l39Bl 3lI39llg inte ehaes Laplace dleeeeidletl ate seelt reassetanee lEiEEWlhE flBii anal sueeeeeled in persisting that the ideal slat sjtsteet ef meatheemeaties is a stable elyiiameieal lS39tEm that wialel enirlute ueehaenged fer all time aehieverlleent was enley eiie ef the letig esetles ef triumphs treeetrdeel it his II39ltJFlllll39 E39 l al treatise M39etenieqee Celeste pehillisfhetil in ve velllmes item N99 to whieh summed up the tit Dtlt en grew39ltartijetl est E wil Flax geeeteatiierlst ef llelustfli ues malthemeeatieieanse Uenfertl1eneatel3r iter his flatet itepultateiieleth he emietted all telerettee the the dl5lU 1rlErliE 5 elf his ptedeeesesers arid E h trlep al aet39iESe aelitl left w t he iitf etreeede thal 0 icleas were entirely his DWI1 Many IlsEEZdtZlllIl35 are asseeiatecle witth this wetlt Dee eatquot the best llme wtt Clesetiihes 08t eeeasiee ea whieh N P leEbDI l tti etl te get a rise eut ef Laplaee hjy pretesetitlg that he had Wfllle e a huge heels tie thee systleettie ell wtlel witheut eitee 1lquott5lEl1lll Illt39eleg Gee as the eaethet elf the E1iViy39iirB iE39 39E39i Leelaee is seiulppesed te heave relied I had its need as that liypethesis g ptlneipall legaey ef the Meeeeiqee Ce le1ste he later gel39tet39eatiiens slay ll39l Laplaeee se whelesal e dieveelepme enquott ef petleenlttfiale El39lEDl jl with its far I39Ee CllquotllIlg iilmmpleieaiti ens let a tlezene CllE EEt Et ll hmrlehese ef phgtstileeaetl seietiee teaitigietlzg ftem grasitatien and held meehatlies te eleette meageneetisnle and aetemle physics Evert theugh he llflfE Cl thee lea ef the petetitlael frem Lagreansgiee witheeet EliI39l WlEvClglTlEll he expleilted V see esttenstiettely tliat eeet sinee P B 2U the lieedealnieneteael d39l EfEit39l lZlal eqeatietl et peteritial the eety has been lCllquotllJW as Laplaeels eqeatlient His ether measterpiieee was the treatise Thames Anelytiqeel dies Preheehfleiteit ll812jIt la ishielh he irleetpetatetl PL we vltlllsee vetsi es iii pttefhahillty ltertt the pteeedithg 40 jteatrs Agalen he f lletll ate aeilteewledge the many ideas ef ethers he mixed 0 G with his ewe hut eseti diseetltitiltlg this his lheielt 9 gEl lEtquotEll agreeeel te he the greatest eetltrihetietl ate this part ef e l jlll lll lel Ihy e zjrl erie titan ll the inttietltletilie n he sayst At bU39llfilC rtI1 the tehesety ell ptquotUlD hllltquot is nely eemrtlee setise rEEl uEEd tee eealeulleateietl Tfltis eta be sea P 6 the feillliewing WU phages ef iiitrieate eaealysiismie which he freely used Laplaee tteansfietmse getletatlrlg fll C LAPLACE TRANSFORMS 413 tions and many other highly nontrivial tools has been said by some to surpass in complexity even the M canique C leste After the French Revolution Laplace s political talents and greed for position came to full ower His countrymen speak ironically of his suppleness and versatility as a politician What this really means is that each time there was a change of regime and there were many Laplace smoothly adapted himself by changing his principles back and forth between fervent republicanism and fawning roya1ism and each time he emerged with a better job and grander titles He has been aptly compared with the apocryphal Vicar of Bray in English literature who was twice a Catholic and twice a Protestant The Vicar is said to have replied as follows to the charge of being a turncoat Not so neither for if I changed my religion I am sure I kept true to my principle which is to live and die the Vicar of Bray To balance his faults Laplace was always generous in giving assistance and encouragement to younger scientists From time to time he helped forward in their careers such men as the chemist GayLussac the traveler and naturalist Humboldt the physicist Poisson and appropriately the young Cauchy who was destined to become one of the chief architects of nineteenth century mathematics APPENDIX B ABEL Niels Henrik Abel 18021829 was one of the foremost mathematicians of the nineteenth century and probably the greatest genius produced by the Scandinavian countries Along with his contemporaries Gauss and a Cauchy Abel was one of the pioneers in the development of modern mathematics which is characterized by its insistence on rigorous proof His career was a poignant blend of goodhumored optimism under the strains of poverty and neglect modest satisfaction in the many towering achievements of his brief maturity and patient resignation in the face of an early death Abel was one of six children in the family of a poor Norwegian country minister His great abilities were recognized and encouraged by one of his teachers when he was only sixteen and soon he was reading and digesting the works of Newton Euler and Lagrange As a comment on this experience he inserted the following marginal remark in one of his later mathematical notebooks It appears to me that if one wants to make progress in mathematics one should study the masters and not the pupils When Abel was only eighteen his father died and left the family destitute They subsisted by the aid of friends and neighbors and somehow the boy helped by contributions from several professors managed to enter the University of Oslo in 1821 His earliest researches 414 eteeteaezma1 EClllATll l lS w39eret published in 1323 and included his aelutlen ef the elasale tauteehrente ptr Dbltet1391 means the integral eqaattien dlaenaaed ln Steetlen was the hrat seletitlen ef an elqealtlten ell this ltlnd and f etrleshaldetwed the eatenalate detereleprnentt elf integrxal teqeatierna in the late nineteenth and nearly ttwenlttiettlh tettentnrlee He alae PTDIVE Cl that the general N degree equa39tlttn nrx the neg edge ex 0 eannet he seflvted in terms ef radicals are peaslhle fer ernatletnte elf lewverl degree and thrua diapeaed el al preblem that had blamed rnathemlaltieilalna fer Sllll years Ellen published his preef in a s parnpahlltet at his men etzaptneel In his aeientl e deaelareprnenttt Ah el aeen entgrew Flerway and lenged te alailtt lFrranete and Grelrnteny 0 the lbaeikitng elf his friends and prefeasetra he appllled tn the geaertnrnaent and alter the tuasnal red 39tapel and delaya he reeeltted at fellrewahtiap fer a trnathelnatieal grand tear ef the Ceintltnenttt He spent rneat el his first year alaread in Berlin Here he the great gene l39ertnne tel malte the aeqnaintatnee ef Aeguat Leepeld Crelle an enthuaiatatltle mathernatieal amateur wfhe heeame his eleae firltend Etil htls f and preteeterl ln ttern Abel tinaplred Crtelle tel llaaneh his larnena Jearnel flirt die Reine and Elegewlnnnlt2et lMethemet39llk wthtieh was the werld a hrat perieldlieal detteted whelly te mathterttattleatl researeh The hrst three aelarnlea ElD1 m ti El Ed EttZlItl39llf39llb ll llill39DlTIS hwy Abel Abelh af llt n1atth ematlealt training had been eaelualaelgyt in the elder aterlnal traditien ef the eigh39t eenth etentetr as typi ed Euler ln Berlin he earned under the tln uelnete ef the new aeheel ef theught led by Glades and Caueht39y whtieh emphlaaieerd riereea dednetfien as eppealtedf te fermlalt e alleetlattl etnr Esteeet fer 39Gtatua5Ta great werlt en the l1p39El39gEA mEIT39lt series there were htatrtdly any preefa in antlalyelta that weuld he aeeepted as teday As Abel etxpreatsedt lt in a letter te friend If jteut dlaregatrtl the vent aln1pl eat eaeea there la in all elf mathernatlea net la ailngle ln nlte laerleel wheee Elj i has beetn rlgertetualy dentermlned lzn ethepr werda the teeat lnnpertant 39393 HE ef ntathematlea stand wl thteet a feendatietn In this parted he arete hie elaaale sftluudyt ef the ilnernial aerlea in whieh he feended the gener5al theerj el eenvetrgelnee and have the hrait aatliafacteryr preef ef the valitdrity el this aerlea etiapanarlenl Abel had sent rte Game in Gdttingen his pannphlet en the degree equatt ltent heplng that it weld aerate aa a latlnd ef aelentlh c paaapertt Helwervetr tear sterner teaser Gauss pet lt laaide WLlIlll1U39UI leeking at it fer it was learned enent ameng his papera after his death years later Unfertunatlzelyt fer hearth men Ahell felt that had heen anablhed and deeided rte ge en te Plarlie wltthleet 39alalting G an5eE ln Paris he met Catlehyt Iegendre Dlrlehtltet and etrheral but these meetings were J39EF f l1 rCllD39f39t and he was net reeegnlaed fer lwhat he was He he already pahliehed a nemfher elf impertlant lartlelea in Crellela Jenreel hat the Frreneh were hardly aware yell ef the etaiatenee ef this new perledleal and Ahead we nnteeh tee shy rte epeala ef hia ewe werlt the LAPLACE TRANSFORMS 415 people he scarcely knew Soon after his arrival he nished his great M moire sur une Propri te G n rale d une Classe Tr s Etendue des Fonctions Transcendantes which he regarded as his masterpiece This work contains the discovery about integrals of algebraic functions now known as Abel s theorem and is the foundation for the later theory of Abelian integrals Abelian functions and much of algebraic geometry Decades later Hermite is said to have remarked of this M moire Abel has left mathematicians enough to keep them busy for 500 years Jacobi described Abel s theorem as the greatest discovery in integral calculus of the nineteenth century Abel submitted his manuscript to the French Academy He hoped that it would bring him to the notice of the French mathematicians but he waited in vain until his purse was empty and he was forced to return to Berlin What happened was this the manuscript was given to Cauchy and Legendre for examination Cauchy took it home mislaid it and forgot all about it and it was not published until 1841 when again the manuscript was lost before the proof sheets were read The original nally turned up in Florence in 19525 In Berlin Abel nished his rst revolutionary article on elliptic functions a subject he had been working on for several years and then went back to Norway deeply in debt He had expected on his return to be appointed to a professorship at the university but once again his hopes were dashed He lived by tutoring and for a brief time held a substitute teaching positon During this period he worked incessantly mainly on the theory of the elliptic functions that he had discovered as the inverses of elliptic integrals This theory quickly took its place as one of the major elds of nineteenth century analysis with many applications to number theory mathematical physics and algebraic geometry Meanwhile Abel s fame had spread to all the mathematical centers of Europe and he stood among the elite of the world s mathematicians but in his isolation he was unaware of it By early 1829 the tuberculosis he contracted on his journey had progressed to the point where he was unable to work and in the spring of that year he died at the age of twentysix As an ironic postcript shortly after his death Crelle wrote that his efforts had been successful and that Abel would be appointed to the chair of mathematics in Berlin Crelle eulogized Abel in his Journal as follows All of Abel s works carry the imprint of an ingenuity and force of thought which is amazing One may say that he was able to penetrate all obstacles down to the very foundation of the problem with a force which appeared 5 For the details of this astonishing story seethe ne book by O Ore Niels Henrik Abel Mathematician Extraordinary University of Minnesota Press Minneapolis 1957 416 mzFFEREmrmL ErUaeTIm4s irresist4ibe 392 HE dis lingAu ishampd him5elVf Equally by the puri ty and n bilit of his rharact 1r1 and by a rare modesty whVichV ma a his person cherighed mm the sam unusaml degree as was his genius Mathematician5 hmwevar have their quotwn ways 0f rememleri ng their great men and 30 we speak uf Abem intampgra l cquatium Abneliarn integra1s and functirms AbElian grImVps Abe1 s aeriqges Ahem partiall Eummati0n formula Ab cl s Iiimift Eheaijrrem in the thencrfy Hf p wer serias and Abel summabili 39ty FEW haure had their names Alinlmczll to 50 many c0nce4ptsV and th UFEmS in mudern rmathematiwss and what he might have a mmpli5hEd in a rmrmal lifaitim is beyond Dnjectuwre CHAPTER SYSTEMS OF FIRST ORDER EQUATIONS 54 GENERAL REMARKS ON SYSTEMS One of the fundamental concepts of analysis is that of a system of n simultaneous rst order differential equations If yx y2x yx are unknown functions of a single independent variable x then the most general system of interest to us is one in which their derivatives yj yg y39 are explicitly given as functions ofx and y yz y Yi f1X 1 2 39 Jyn Y2 1 f2x9y l2y22 2 yr yr fnxgtyly2 39 39 2 Systems of differential equations arise quite naturally in many scienti c problems In Section 22 we used a system of two second order linear equations to describe the motion of coupled harmonic oscillators in the example below we shall see how they occur in connection with dynamical systems having several degrees of freedom and in Section 57 we will use them to analyze a simple biological community composed of different Species of animals interacting with one another 417 DHIFFER EN3939l a L EQUHTTi39U39NE An imperteam ma themat iAeaaI reaeaH for studying systems is that the eirmgle ath DIFUET equatiIn1 W f xrm as y 2 can aalwaaye be regarded ae a special ease ef 1 Ten see this we put J J Pa J ya QVMJH and ebeewe that 2 is equaiva1eaenat tn the eyatem ai y Z fx39ylry239 39 quot 39WIhiGh is clearly a special case af q Thee statement that and 4 are e quiavaalaem is anderataeeda he mean the fnllewalngz if yt is a eelutien af equatien 2 then the f uaeatiena yIx y2x e 3 Mix clae ned by 3 aatiefy 4 and eanverselieya if ylmr y2r x yaa satisfy 4 then yfx yi xj Via a eelutian af 2 This reduetien af ans mm aredear equamiaan 10 a 3etexa af a rat arder eqaatiene has several advanta geaa We Hixustraeate bye eenaideariang the re a mien haetweern the baaie existence and uniqL1eneaS the f m far the eystem 1 and fear eqaatiaan 2 If a xed paint AI M is eheaen and the valauea ef the aanalwanawn fuaeatiena A Lp 113 0Ak E 39 39 quot 1 yurl xliil u m are assigned fI3Ji1Zf3I iI in such a way thaaft the funetiena j I a are de ned then 1 gives the vaiuees wf the dee39rivativee yi xnfl y xUa 0th I The ai1nilarity b etweea thie aituxatiaan and that die eaased in Seetien 2 eI1ggea1s the fel awinge analog ef Pieearefe th enrem Thenrem Let the fanzerfana va P a39ad the par Emquot deri39aeriaea BfTfampy J fef393yM xi 1 539 fe y 1i f f yn be eaeerfneaaara in a regiyen R erf ux ya 1 y apa eeE If Iquotma aA 39u an W an 39aI39eerier pain af R then the ayarem 1 has a anfqame aaaI39ean ynfa y2c yr me 5a39a39a ea the fairies eead 39 i0r1a Q We will not preve this theorem hm inaieead remark that when the g1quotI3l1l dl has been preperhr preparaeade its pmaf is identical with that of Pquotieardquote thearetmn as giaea in Chapter 13 Furatelmrmere by virtue if the aheve re daetiane Tfmeeeem A irne1ude as a speeiea4I ease the fe ewianlg eer raeaspaneding theareem fer aeaaatiaane MF SYSTEMS OF FIRST ORDER EQUATIONS 419 Theorem B Let the function f and the partial derivatives 8fBy 8f6y 8f8y quot39 be continuous in a region R of xyy yquot39quot space If x0aa2 a is an interior point of R then equation 2 has a unique solution yx that satis es the initial conditions yx a y39x a2 yquotquot x a As a further illustration of the value of reducing higher order equations to systems of rst order equations we consider the famous nbody problem of classical mechanics Let n particles with masses my be located at points xy2 and assume that they attract one another according to Newton s law of gravitation If r is the distance between m and m and if 6 is the angle from the positive xiaxis to the segment joining them Fig 63 then the x component of the force exerted on m by m is Gfnimj G7n quot1jxj quot x y S 2 3 V1 ij where G is the gravitational constant Since the sum of these components for all ja i equals mdzxdtz we have n second order differential equa ons d2x m x x 1 G 2 dtz j 39 77 and similarly d2 I G2L 2 1 he ril zn mi I r m 9 X HGURE63 j TDIFTquotEREHTIAL EDUHFIAHHE and h 2 G 0Y rwirs 3 d N If we put 145 g uh dyJd and 39fJ 3 df4d r and apply the allimve reductimn than we Dtain 3 Eys39teVm Tiff r Eq39uatinns of Ih farm 1 in lilfjne unknUw1n functi n5 Jr quotMW g P e xn via ya xum 0 u yT uh Eh isrw d 3 win If WE new mak use of the Eacst that b rm a e 5a v3 a men Tha rewm wields ma f0llw4ing cmn cl39u5inn if the iinilia I p0 siti ns and initiaI valoc ities mf Pt gpamicllues 0P the Vvalru es mt ma Lmkn wn functions at a certain insmm I rm am gjiquotFEITi and U this particfres do nnt mH39idi E in the sense Ihatj the Jr d not vaniahi than Eh iiiifif su5eq4Vuent pnaitimns and vEh JCitiES am uVniquEljr determinEdJ p L CUmIusim1 un derlies the once popular p hAil03Dphj Uf mechamisti deterVn1inisVm award ing in which the uxniverse is nwhing rnme than a giantic machinE whose futurE is inexDraMy xed by its state at am given m nmentil PR M Replace eaeh f the fn2MnAw39ing diff rntia equata71m5 xI an equivalenI 5stem Df rs Ird Er Equatisunaz W WT E 2 G If 0r partiI of ma a m moms in lathe my1 pane4 its equmi ns of mrUtimvn am mE 2 1 D Jquot and E F 1 39 K 9 39I W gt H W39hi39 F f and D repVresaent rm I andl y Acmmjponaatnta re pectivel39 mf tha fm39ce awaiting am the panjiIe Replime thj5 SquotEJEIn 7i mf irwn sasmmd arr Jar Eqj39lJ IiU m5 by an equiva crmt sjygtem of fm1r rist nrder eqMations emf the Afarirn 21 quotIt amu Ed Sir amEs Jeans to dz Tthtr 11i i39a39 39I SE as 393 smlf5a s l ving jf5tEm nf EN 3imuI39Eanemus diff ErE mia Equamfi0n5 wLhcre N has Eddinglun s ziuniber Sir J5mih ucr Et dijng mrn assermd wi39th mam pIn39iry39 39than trut 1 that N 12Tc 13 3 25 2 the wI rtaW n39umber39 Bf partirzlhe of maMer in the L1139I39ivEirsnE Sew Jeansi W1 E 4 fF FI HIi E7HI39A39 H rimn CJ391 fDr UI39Iiwr5it Pr r LIIznE1 I 1i DI EE1 ii39ngIl uvn THE Expanrgiirlg UHEU r3E zCan1l1r39iJgamp UniIvers139 Lnnrdm 1 SYSTEMS OF FIRST ORDER EQUATIONS 421 55 LINEAR SYSTEMS For the sake of convenience and clarity we restrict our attention through the rest of this chapter to systems of only two rst order equations in two unknown functions of the form d 2 Ftxy 1 dy t dt Gtxy The brace notation is used to emphasize the fact that the equations are linked together and the choice of the letter t for the independent variable and x and y for the dependent variables is customary in this case for reasons that will appear later In this and the next section we specialize even further to linear systems of the form amubm m E1 139 a2 x l 1720 4 f2t 2 We shall assume in the present discussion and in the theorems stated below that the functions at bt and ft i 1 2 are continuous on a certain closed interval ab of the taxis If ft and f2t are identically zero then the system 2 is called homogeneous otherwise it is said to be nonhomogeneous A solution of 2 on ab is of course a pair of functions xt and yt that satisfy both equations of 2 throughout this interval We shall write such a solution in the form x xt y yt Thus it is easy to verify that the homogeneous linear system with constant coef cients 4x y W 0 C 2x has both 3 am Zr lj Z3 and l 82 W as solutions on any closed interval 0 DIFFEElEHTiAL iEQL r Ft WS We new give a htief shetieh ef the generals itsheety ef the iineer sytsitemi e It will he e hseLrvemi that this thieerr is watery similsr te that ef the seeend etder lineer eqiuetisten as deserihed in Seetiens 14 end 15 We begin by sitetieg the tieiitewiini ifLmdsmetntsl e sitienee and 39U iqUEt1ES5 thseereitn wheste preef is given in Ctheptiet 13 Theerem d ifs is easy Apetinst ef the inrereeii ash end iFKti and En ere I39aPI39 numbers PHi1 Jii U EF titers 0z hes entie end ere ee seitttfetn Ar rrI sMm te ia mretugheur ehis sueh that xiri t eerie jr39ifg it y Ger next step tel study the stme391tu e ef the selutieits ei the thernegeneeus system ehtsined item x rtemetving the terms j39i j sine p u i is Eeamebme amnbme 5 it is eihivitelus that p is ssrtis ed by the seaeaiied triuisi seht ien in w39hieh mm and yew ere berth idiientieieiIiii3r EEi39 Qur mains teeii iin E 539t Eli39IquotIg mere useifui seiutiens is the next theeremg Theieretrn P o f the the srtegerteaem sys39 em g hes hm seii39etieIquotHtT Jr 39t and At xii suit i F pa 2 iTi en mph thee iiit 5 139uii 39E139139eitii39 132 39 Cu39Iiii i quot39 wtzI J is eise ls stehHien en eibi fer hey eeestiemfs 21 and E3 Preefi The prieei is s teutine seri estien and hiss left tie the reader The selutien 7 is ehtsined item the pair ef seiiutieesi 62 by imutlitiipiyinilg the rst shy in the sieeend by tgj and eddjihg 393 is therefere ealiiedi s lr irI E 39nF eemhtirwriee ei the selutiens With PI tesrminseiegy we we restate Theorem as fellows any linear eemhiiinetiieini ef tiwe sieluitiens ef the herrne39gegtn eie LIsi S SJiEI 7t I is else is selutieae Ae1cerdingLiy 3 has 3 st site eye P V 34 T 2 y e site 2523 es s sehltLtien fer i h ifji eheiee ef the eemstents em and Egg SYSTEMS OF FIRST ORDER EQUATIONS J The next question we must settle is that of whether 7 contains all solutions of 5 on ab that is whether it is the general solution of 5 on ab By Theorem A 7 will be the general solution if the constants c and C2 can be chosen so as to satisfy arbitrary conditions xt x and yt yo at an arbitrary point to in ab or equivalently if the system of linear algebraic equations Cv 39i ti L C2x2 u X0 Clyt C2 2tu Yo in the unknowns c and C2 can be solved for each t in ab and every pair of numbers x and y By the elementary theory of determinants this is possible whenever the determinant of the coef cients 160 x20 MU y2f does not vanish on the interval ab This determinant is called the Wronskian of the two solutions 6 see Problem 4 and the above remarks prove the next theorem Wt 3 Theorem C If the two solutions 6 of the homogeneous system 5 have a Wronskian Wt that does not vanish on ab then 7 is the general solution of 5 on this interval It follows from this theorem that 8 is the general solution of 3 on any closed interval for the Wronskian of the two solutions 4 is 63 e2r 63 262 Wt es which never vanishes It is useful to know as this example suggests that the vanishing or nonvanishing of the Wronskian Wt of two solutions does not depend on the choice of t To state it formally we have Theorem D If Wt is the Wronskian of the two solutions 6 of the homogeneous system 5 then Wt is either identically zero or nowhere zero on a b Proof A simple calculation shows that Wt satis es the rst order differential equation dW 7 am bzmlw 9 from which it follows that Wt Cell l39h2 l 10 for some constant c The conclusion of the theorem is now evident from the fact that the exponential factor in 10 never vanishes on ab 412 It e1tFeEneeTnt EQUATID N5 Thtedrteim C prentidee an adequate mteans ef ver39ifying that T in the general eelutinn et 5 site that the Wrnnekien b nf the tum eeriutidne ddee nt venieh We new deveinap nil equivalent test that its Dften mme direct and cenveniiietntii The twe selutiene 6 are eellet iinenriy dependent on me if one is e ennstnniit tnmlttijptlte ef the ether in the sense that p7 kxzfii M xiii yitt team Jet knit fer senile constant it end nil I in ten end itineetriy independent neiither is en eeinstnnt ntulttiplte 0f the ether It is eteer that liineer dependence eqiuiveienit tn the eenditintn that there Btti i twee eenetente he and eg at tenet ended inf whiten is net zero such that PB 0 pus E x quot3931 Jhiii quot39 Cziyzm 3 fer nl E in eb We netw have the next thee139ern 11 The rem PG if the trwe eefeirierte 6 er the hemegeeeeue System are iieeeri39y ind39epe39nd39ent en eb then T is the genueref eet39nfen ef en ithie ieter39uet39 Preef in View nf Theererns C and D it suffieee 39IfiI1IfI39 shew thet the snint39iene 6 ere iineeriy dependent if end entity if their Wrdne1ziian W13 is identtieaih eere We haeigin 39Iw aeeumitng that tiejr are iineerly dependenm ee theti eeyr t Int IaW gtiquot13910 xI EI quotIii 7k r ei3quot39t fair tiIzifiyatt iiI2if 2 FJ 3 int nil It in The S tfntl ergnmetnt weeks equality well if the eeine39tnnt R is en the ether side of equetien5 12 We new assume that WU is identieeily eere end ehew tthet the selntiens 6 are lineeriliy dependent in the eenee elf evqne39tiadns j Let I be 3 xed peint in eh Eiinee E the eyetent ef linear elebrnie equatiens 12 Then Wm cxx t1r2ti U i7iL Iii ittii quot3 2i iiiii D has n E39ti39Lt39tiidn e E3 in whieh theee nnrnhiers at e net berth zere quotThus the eeltitidin et 5 given by Lt egJti t e2r2 t Jr Eiyiiij ii quot 32 2ii 13 L1 SYSTEMS OF FIRST ORDER EQUATIONS equals the trivial solution at to It now follows from the uniqueness part of Theorem A that 13 must equal the trivial solution throughout the interval ab so 11 holds and the proof is complete The value of this test is that in speci c problems it is usually a simple matter of inspection to decide whether two solutions of 5 are linearly independent or not We now return to the nonhomogeneous system 2 and conclude our discussion with Theorem F If the two solutions 6 of the homogeneous system 5 are linearly independent on ab and if xpt y ypt is any particular solution of 2 on this interval then x cxt cx2t xt 14 Y Clylt C2 2t pt is the general solution of 2 on ab Proof It suffices to show that if X xt y yt is an arbitrary solution of 2 then X xt xgt0 y Y0 ypt is a solution of 5 and this we leave to the reader The above treatment of the linear system 2 shows how its general solution 14 can be built up out of simpler pieces But how do we nd these pieces Unfortunately as in the case of second order linear equations there does not exist any general method that always works In the next section we discuss an important special case in which this problem can be solved that in which the coefficients at and bt i 1 2 are constants PROBLEMS 1 Prove Theorem B 2 Finish the proof of Theorem F 3 Verify equation 9 426 B39llFFERENTlAIL EtaUhmmzs 4 Let the SKECTUIM DrdE39 liinh1r hquatihn Fr L I in K PfNE was 2 0 W rezducehd ID the Equot 39EEm I an V V 7 J M fili hm Pma If MEI and 139r are s luftihns of equa Ihinn A39 and p2 H S P F yTr J and P Agift are the EZCIETEEPDlldiiflg scznlutions f 39 Sh w thagt the quotwmnskianh of the farmer in hlthe sans Inf Semi n 15 is pmzisghr the Wr nslszian hf tghfhe l3 ttvE l in the 5Esme mf this ectiU nh 3 Shaw tfhm E1I h F P and 0 pl 4 quot 39w 3 M 5 H N is are 5nluthihnn5 If tlh hhumUgenenus sysxthem AM Slhw in twu ways that the given slu1i n5 if the sfysterm in 3 are inearyh indepEndehn1t an Evh139y clnsedi iintEwa1h and wrim the gencrai S llllii n hf 0L 539yhtehmh I P5 Jr W I Fiind the particu ahr Shlquot1IIhlMiEIIquotI f this systam for whhmh x S and hy 1 fa Slmwtha1 N I W Jr 25 pix 539 aid 3 W ly SE arhe 5lutinh5 of the hcm0g nmhuh s35tezm A d I V dy SYSTEMS OF FIRST ORDER EQUATIONS b Show in two ways that the given solutions of the system in a are linearly independent on every closed interval and write the general solution of this system c Show that X 3r 2 y 2 3 is a particular solution of the nonhomogeneous system dx Ex2yt 1 3x2y512 and write the general solution of this system 7 Obtain the given solutions of the homogeneous system in Problem 6 a by differentiating the rst equation with respect to t and eliminating y b by differentiating the second equation with respect to t and eliminating x 8 Use a method suggested by Problem 7 to nd the general solution of the system quot x dt y dy dt y 9 a Find the general solution of the system dx iquott39 X dy EY b Show that any second order equation obtained from the system in a is not equivalent to this system in the sense that it has solutions that are not part of any solution of the system Thus although higher order equations are equivalent to systems the reverse is not true and systems are more generaL 56 HOMOGENEOUS LINEAR SYSTEMS WITH CONSTANT COEFFICIENTS We are now in a position to give a complete explicit solution of the simple system dx ax by dt 1 1 C12 C12I bzy p DiFFERENTML EDUATIUNS wiheare 11 hi 1 H3 alri1Ib3 art givan mn5itantsE S m 0f the pZF iiJiiEm5 at 39Ill39IE end of me pr39EViG HS semiown iiilJjIE iZIfv31 E a pgrucedure that can nften be apgpliied In this IEBLEE di errientiiaitei one equiatii mn elimiiiniata ne afi the 39IZiEfpE d i i variaihle5i an sufliwf resulitiingi SE m m DTI iELI iiin ar Equa timL Th P we imw dEs lfib is base i1i5t Ead ninn5truictiiing a pair iinieairlyr iiidcpcniidient 50l U39lIim39it5 idiirieictijir fmm the giiveni sysitem if we retail that the Expnci1tiai fuiniucti n has Eh jpre pmity that its deriiviaitiivgsi am r1nstam multiples nf the ifunctii u1 ittgeif tihein just as in Sactiitm 7139 is El1ILI1l39 i in seEk SifJ39IlL139iiZii1BT15 mi N Lha39viin the form J 5 iAequotquotquot V i y BE If we substitute 2 inm 1 we get Aimem mA b1e quot me gAE quotquot bgBie and diividini by ism yields the quotlirmawr ialgehiraiic systtm v i F 3 in the Haiku3 wins and It is II iEET tihaiti 3 ifnas the triiviai soiutiion P g B E Di iwhivziii miakes E the triviiai SiUiiUili n of IL SirltE we are ii U39iiiT1g for nanxtrivial simlmi ns Inf 1 this is nu hefl p at alli HGWEVEI we know that 3 has 11t1L1iI1Ii i s39iL i smliutii nisi wihieniever the dertarminiant 0f the 39E EE39iC iE li5 vamisihrasi 0 whrnevEr ij 33939 hi 1 2 0 When determiniianit is epandedi we get Ih quadiratic eiquati n far that i k 39WIH o anaiggg with mir pFE Vi UE iwmk WE call thiis the aiuxiiiuiry Equm39imni 0f the system LEI NI1 amid he hie rats of 4 If we rapiam in 3 by p theni we kn w that the IEsuitiing E2ql1 39tisD S have a nimtriviiiial so lu1im1 A1 B1 30 I A mquotIi is a innn st riivial sniutimi of til system H By PFDtEE5E di g simiiparly wiiith m43 WE nd anmhcr 139m ntI i39vial s iuitiim x 0O 2 Hzg flglm SYSTEMS or FIRST ORDER EQUATIONS 429 In order to make sure that we obtain two linearly independent solutions and hence the general solution it is necessary to examine in detail each of the three possibilities for m and mz Distinct real roots When m and quot12 are distinct real numbers then 5 and 6 are easily Seen to be linearly independent why and X C1A1emlt C2A2 m239 F71 7712 y c1Be c2B2e is the general solution of 1 Example 1 In the case of the system 5 x dt y 8 dy 4 2 lt gt x dt y 3 is 1 mA B 0 9 4A 2 mB 0 The auxiliary equation here is m2m60 or m3m 20 so m and mg are 3 and 2 With m 3 9 becomes 4A B 0 4A B 0 A Simple nontrivial solution of this System is A 1 B 4 so we have x 83 10 y 4e3 lt gt as a nontrivial solution of 8 With m 2 9 becomes A B 0 4A 4B 0 and a simple nontrivial solution is A 1 B 1 This yields x 82 11 y lt gt as another solution of 8 and Since it is clear that 10 and 11 are linearly independent 12 x ce 3 czez y 4cequot3 C262 is the general solution of 8 430 ii IFFEHE f TI39ML nEnn UAr1nH5 Dnisstimt un mpplex rnms If mg and ring are disftlirnnct cnpnmp nxn nnnmberns than they can be written c Eh farm n i 55 wnnhnere 1 and b are real I39Hili IlbuET139quot5 and b 5 S In this casna we Expect the A3 and H5 n hvtnanin Bdn from 3 I01 tan C inipl t En nmbman and we nlquotiETl u39Ei two linearhs nindependnepnnt snnlminnsn x ArEInEbr and I AEn f I39 y BrE m b39I B39ea br n nnwever these are cUmpVIexvalnu edf snInIni nns and In extracnt reanI nvnln Ed snilnntinns WE prnncneaedl as nfn nwn P WE enxpress the HUWEEIIS A T and 1quot in the standard fnrrn AI 2 A EA and f 3 EH3 393nd u r Euler39s f39DF M 1 7 than the rst of the snlnti nns 13 can be written an 1 Ag EA2Equot cnsn P sin bx J E BE c39 2E ens bi Enin bi nr ix e A cans bf VA sin bf fish sin bi A2cnsbr M 1 y E WMB cnnsbf 3 5111 EM Bs1nn bi 52 ms b39I It easy tn me that if a pair of mnmplexnwaiundn funee1innnL5 is n snlutinn nraf 1 in whinh the non iiciients are ran cnnstnant5 than thEir twn real part5 and Nthneir twn ninmaginnnry pnm are rrealnvalued BallIut iD I E It fflnlows from o that M Fi ldll the twn meanwaluendn soiuti ns I 9 1e quotIA Ens bf VA sin EN 15 y E e E con bf 393 sin bi and I e Fsi15in bi Ag C U5 br 16 P e 39397Bi 5 Lb Bcn5EvI7 n It can be shnwn that these swnnluntninns are lfil1E1 1quothf39 indnependEnnt we ask the rnnader tin PIQVE mk in F mbil39n1 qj sun the gE39lquotI E 1T a l 5n luntni nn in CREE is I e cm AE can 6i c sin bi cgA1s in hr Ag ens hm HT 391 e c391nB nnsb Basin bi c 2B Si bf E 39S b c Since we hnnve alraadny fnnnd the genemfl soflntinnn is new irIEC E5 quot39 In nnn5indenr Ih secand f the twn Lsnlutfi ns Efqnnl real Innis Whpen rm and hnawe me same value than 5 and 6 are not ilninaajrly indEpEnd ent and were esnsnntinlly ihnve nnly one 5nlri1ntiDnn J As H 13 L X X Bequot Wnr EpEfiBTfI in Scctinn 17 would lead us In ezzsznpent 2 sneond 1inear1ny SYSTEMS or FIRST ORDER EQUATIONS 431 independent solution of the form x Atequot y Bte Unfortunately the matter is not quite as simple as this and we must actually look for a second solution of the form x A A2tequotquot 19 Y B1 3209quot so that the general solution is x c1Aequot c2A A2tequotquot 20 y 2 c1Be c2B B tequotquot2 The constants A1 A2 B1 and B2 are found by substituting 19 into the system 1 Instead of trying to carry this through in the general case we illustrate the method by showing how it works in a simple example Example 2 In the case of the system dx 2 3x 4y d 21 Y 2 2 y 3 is 3 mA 4B O 22 A1mB0 The auxiliary equation is m2 2m1O or m 12O 2The only exception to this statement occurs when a b2 a and a2 b 0 so that the auxiliary equation is m2 2am a2 0 m a and the constants A and B in 18 are completely unrestricted In this case the general solution of 1 is obviously X cequotquot y Czemr and the system is said to be uncoupled since each equation can be solved independently of the other P DEFFER tiTIALE fltT1EZl H5 whquotiEh has aqua real rszzamts I and 1 With m 1 22 betsom s A 28 L A simple tn0ntriviaI sui39u ti t1 if this system is A i 2 539at tr 26 7 is a D39nHquotiiVLi al snltut nn of 21 We new set3k 3 53 mud HI39ftE394 I 39Iju39 amp39ndEpendent s tluti n Df 39thE ftJtm I A Aztyet W y 173 B1tgquot 3941i hEtt this is s ub tit utEd iirntm WE obtaitt A 213 A1equot 3m Aztjzt E tw B 4 Bit tgjie E 4 A2tEquot B 1 E which radtutms at once tn EA t 43 EA A 43 lt D A3 Z t A EB Srinwc these are i39EI39t be identities in the variabl t we irmztst hmre E 43 EA A E 43 D A EB D A r E B The tum eqruatjnns mm the Ief39t hatelt A P 2 H 1 as a timpie nontrrivial 5nIlut i mn II39Ei5 the twat eqAuati 39rts U11 the right hECDfn IE 24 43 2 E PE 11 S6 we may taIke A 1 B D we mmr it1sEtt th E5 E t1 I39Il i v7D 39E inti If amt tj bvtain I t 1 2 et39 quot E are as mIr ztewndt sntutmnt It w Dbquottquoti lJ5 tthat 23 and T2 t am lin5atl It E39 Etdjn I 50 25 5 PO 2t 1 l 2tte quot26 39 1 E t1tE39 E is the gengtal 50utim1 of the 53rgtem If2tl 433 SYSTEMS OF FIRST ORDER EQUATIONS PROBLEMS 1 Use the methods described in this section to nd the general solution of each of the following systems f fd dI 3x 4y 5 2x dt J dt altd e 1 u6m Q 3m cdt rdt Vdx Fdx 4x2y 4xy 1 dt b 4 f lt C X5x2y x2y dt id d dx 5x4y 7x6y dt dt 4 Q4 dy xy 4X2x6y dt L t dt f f d Ed4x 3y x2y d J dy h lt 8x 6 4 d y Ldt x 5y Show that the condition azb gt 0 is suf cient but not necessary for the system 1 to have two realvalued linearly independent solutions of the form 2 Show that the Wronskian of the two solutions 15 and 16 is given by W AIB2 A2Bi 2m and prove that AB3 A38 0 Show that in formula 20 the constants A2 and B3 satisfy the same linear algebraic system as the constants A and B and that consequently we may put A2 A and B3 B without any loss of generality Consider the nonhomogeneous linear system amkbm nm d n Y E aztx b2Iy f2t and the corresponding homogeneous system dx atx bty dt H W dt a3tx b2ty 434 DIE E7 EREHTLisL EeusTIess Es If J T i V 7 I 2 W3 and I P ymm yhm are linessrly d E P3I39idE 1i sssehsteiemsz sef s se that L35 5quotQEIUJ 5iIs r J h 4 3e 2I Rm its gensersE seleisiens sheets that 1 j h y uMeMwmeM will be e reameulars se1sustien ef d if the functisens em and vim sstisffy the sysmm liili vi 312 13 13quot39s This t eehmisqu e es nding psr39tieulsr selutinsss ef menhsemegeneseus tinease syslierms is eeHeds the meshed of 1Ier139eiLee sf peremesers Er Apples the method e ut1inesris in a te nd E pameule re selu Men ef the nuemenhseemegeneeuss system H V Ji 39 39 kO V dy gt a 4 V 2quot A 3 3 6 I y I whsesse eJs39rsespendi11g sh en1eeneeues sy39s Eem is sewed in Eaempl e E P G xS Evsesysenet kn ws that tsheres is es etmstsnt struggle fer smvxwse emeng di feresnt speci es ef animals slisvsing in the same e1wir enment One kind If animal e quotvquoti39ll1 Iaquotiquot39ie I 5 by eating ansestsher se sseceemd Aby dlevelepizng rneethseds ef sevesien to 3V Ed being e atem sand en en is simples esemples ef this usnieerssle een iet bestween the predsmr and its prey let us iIlTl Ti 39E an island inhabited by fesees and rabits The fees eat lquot bhsi5 and the 139al3rbIiI S est elevers We assume that tllese is see mueh efeser ertzhst the rebbits eIsewe4s have an ample suppiys ef fessesis When the resbibeists are ebun dsn then the feses C11J1 iEh and their pepus stien grows When the faxes became tee nusmierus and eat we mean srsebshiitese enter a gpesr39ieeI of famine and their p puiE1tiUn begins to deefline As the fezes decrease the ireibbjierts beeexme relseteeieehr safe end their pepulss stien starts Ire sirncresse esgein This trisggeI39s s new iinszreese in the fee pepuelstiem end szs time gees en we see an enec eessly repesetesd cyeele ef SYSTEMS OF FIRST ORDER EQUATIONS b interrelated increases and decreases in the populations of the two species These uctuations are represented graphically in Fig 64 where the sizes of the populations are plotted against time Problems of this kind have been studied by both mathematicians and biologists and it is quite interesting to see how the mathematical conclusions we shall develop con rm and extend the intuitive ideas arrived at in the preceding paragraph In discussing the interaction between the foxes and the rabbits we shall follow the approach of Volterra who initiated the quantitative treatment of such problems If x is the number of rabbits at time t then we should have dx ax agt0 dt as a consequence of the unlimited supply of clover if the number y of foxes is zero It is natural to assume that the number of encounters per 1 ll lr FIGURE 64 3Vito Volterra 18601940 was an eminent ltalian mathematician His early work on integral equations together with that of Fredholrn and Hilbert began the fullscale development of linear analysis that dominated so much of mathematics during the rst half of the twentieth century His vigorous excursions in later life into mathematical biology enriched both mathematics and biology For further details see his Lepons sur la th orie math matique de la latte pour la vie GauthierVillars Paris 1931 or A J Lotka Elements of Mathematical Biology pp 8894 Dover New York 1956 A modern discussion with the Hudson s Bay Company data on the numbers of lynx and hares in Canada from 1847 to 1903 can be found in E R Leigh The Ecological Role of Volterra s Equations in Some Mathematical Problems in Biology American Mathematical Society Providence RI 1968 0LA rwEFFEEEH 1TM E imT1uHs unit it 1 E between rabbit5 and faxes is jnintly pmpnrtionai lit 2 and 0 8 If we fLIIquot39IhEr assumt that at certain prmpnrtmm Uf lth1E SE E cD llI139lEIquotS result in 3 rabbit being t thi we have dx 7 0 1 ax hwy a and b 6 U In the sarnic way l 632 1 i and d DI zI far in tht absmm of rH39bbin3 lzhe f xes die out and their immase dep nds on thc rmmhtr at their En cnunter5 with Iquot l39IJ i539 1ilI We thcArjemrv have the IEnlowing rmnlinear sivstem describing the intueratztinm Df th two Sp iiESI I V d yc dx E quatim15 1 are c gd V quotIeLrmquot3 preypred5amr eq mrmn52 U nif0rtunat Ely39 this system anna1t he s0vecl in terms of elementa4r1v fumrcftimmaa On D t1391ar lmnzd if we mink nf its uznjknaawn s lVutii0n I r tJ 393 Hf as cVnVn5IiJtutiVng the p aramEtri eq ua7Ii nns atquot 3 curve in the 4ry plam E than we can nrlf lfh rectang39ula1r e quatinn nf this ruwe On e4limina ting 1 in 1 Ely divisim1 and S39EP F ti g the vtariabkles we ubtain H by p y C I39r y 1 Integratin3n mw yields Nagy by 2 ac mgx it P 01quot A 39 EEb F Kx 1E whrE1quote cm1stam 0 is giv by f 2 P xampn b u in terms 0f the iinitiail wings of J and A1t39hmugh we rsamn t salve E for ei1ther 2 CHquot we an detnermine pnrinm2s on p rurve by an ingenimmr metfhmd due to VmJlmrra To dm we equate i etft and right sidezss mi 2 to new variables 2quot and W and SYSTEMS or FIRST ORDER EQUATIONS 437 then plot the graphs C and C2 of the functions 2 y equot and w Kx equotquot 3 as shown in Fig 65 Since z w we are confined in the third quadrant to the dotted line L To the maximum value of 2 given by the point A on C1 there corresponds one y and via M on L and the corresponding points A and A on C2 two x s and these determine the bounds between which x may vary Similarly the minimum value of w given by B on C2 leads to N on L and hence to B and B on C1 and these points determine the bounds for y In this way we nd the points P1 P2 and Q1 Q2 on the desired curve C3 Additional points are easily found by starting on L at a point R anywhere between M and N and projecting up to C 1 and over to C3 and then over to C2 and up to C3 as indicated in Fig 65 It is clear that changing the value of K raises or lowers the point B and this expands or contracts the curve C3 Accordingly when K is given various values we obtain a family of ovals about the point S which is all there is of C3 when the minimum value of w equals the maximum value of z We next show that as t increases the corresponding point x y on C3 moves around the curve in a counterclockwise direction To see this FIGURE 65 433 DEFFEREHTIAI EI3UA TI HS we b giin by n vting that tquaItiUns 1 give the hmmrizUmal and vmquotticai compnmcmts of the v el ocity f th5i5 pint simp1c caEecu1ati n DaasedT on fUrmulas 3 Shuws t hat Ilia pnmt i has cncrrdinates Jr 3 c d y when Jr 394 397d it fll0Vws fmm the seamd equatinn mrf 1 that dyfdr PW negativE 50 wt pavzwint an mmres dawn as tr39 quotwer5E5 the arc QEPE Q SJimilar413 moves tip al ng the arc P Qg so the a5sertim is jpnzwed Fi naHr we use the ftrxsrabit pr0bl em tn iIlVustra1e the ifmpnrtant rrzerhmzd VinearizmEmg Firwsm quotWE mbsewe tghat if the rahbit and fax pnDpuiat irUrn are t and 4 I d y b J than the system 1 is satis c and we have dgrmr E and d yfdj 0 so th f are nu incmas4es or deVcLreasEs in I or Vpnp11latimls Q are called equi ibrEum pUpgufm ian5 fun 1 and y can maintam 1hEm5elfveA5 Vi ndE nite4ly at these mn5taAnt Mevels It is ohvisaus that is B 4special rase in whiVrh the minimum of w Eq uaIs maxim um Of 3 50 that the ma C3 red uces In the point If we rmw remrn to the general case and put 1 V 1 1 i P and d 2 Y 1 J b than R and can be thought 0f 35 the deviaIiUns of x and y Lfmm Ih if eq uiJlibriVum values An E S 39 c1alcuat4ian shows that if A and in 1 are r3pVlaCed7i by q and o s ammixms to tran51ating the point If5e39quot Mb In the rwriAgiru than 11 Tb acnmes A Miquot be V W y 2 2 Y dig d cf s L i 0 y air We nnw linearize by asisumyigng that if q and 1 art 5malL men the terms in IE5 can be disaa rded withDut SE1 i US aerrlyrxz This assmmpvti n ammnnts to Httleiz mmE than a Imp hurt it dogs simvp lify 5 to a linsmlwr system It is eagy tg nd the general 5aIutVinn miquot 6 hm it is evenV easier 10 el imEnatE Wk dEvi5inV and obtain cf c1 392 h ax 51a 1 L SYSTEMS or FIRST ORDER EQUATIONS 439 whose solution is immediately seen to be ad2X2 b2cY739 C2 This is a family of ellipses surrounding the origin in the X Y plane Since ellipses are qualitatively similar to the ovals of Fig 65 we have reasonable grounds for hoping that 6 is an acceptable approximation to 5 F We trust that the reader agrees that the fox rabbit problem is interesting for its own sake Beyond this however we have come to appreciate the fact that nonlinear systems present us with problems of a different nature from those we have considered before In studying a system like 1 we have learned to direct our attention to the behavior of solutions near points in the xyplane at which the right sides both vanish we have seen why periodic solutions ie those that yield simple closed curves like C3 in Fig 65 are important and desirable and we have a hint of a method for studying nonlinear systems by means of linear systems that approximate them In the next chapter we shall study nonlinear systems more fully and each of these themes will be worked out in greater detail and generality PROBLEMS 1 Eliminate y from the system 1 and obtain the nonlinear second order equation satis ed by the function xt 2 Show that dzydtz gt 0 whenever dxdt gt 0 What is the meaning of this result in terms of Fig 64 E UATI NS 5 i i E s Y p3 There have been t39wn rmneejnr trnennde in the hietneiennlin d lf ii pm nt of ti9i eeenIinl equnetinnne The urst end nleleert is ehnreeteu39inzed by e emnte In nd enpJieit enlultinnnne either in eelnsned formwhieh is rarely peeible nre in terms nf pewer series In VhE39 5eennnL nne ebennnnne all nhnnpe ef snlving eque tinene5n in en n39ndit innnE sense and inslend reeneenljrenlee en 31 eeereh fer qnnnlietntiee infnrmnetinn nbnu I the genernnl behevinr nf enlntinne We nppiriedi Ithi pninntn nf view in linear eqneteinnnes in Chapter 4 The qJnnniitennve tnenry nf nenlinenr eqneti nnns is tneten y di enrnen L It was fnnnndned 113 Pninenre amend 1880 in enenneetinnn with hie week in ee eetial meehn iC end einee that time lhns been the nhjeeiz nf eteedi y increasing ninnterestn en the pen eff bnth pure and nppli en mnthenme39leieienequot The etheenry nf linear diEerentie1 eqnnnti nne hes been SI39lr1d39 E Efl deep39ls39 nnd extensively fer the past 200 yenre nmi is a fairlyquot enmplete end w elI nrnun ded bIJd7 ef knnwiedgen Hn ewnevere very little eat a general I See Apeniiex fer a genernil eeeernrn emf Fneineeuequote were in minhenmniee E eeieneee E NONLINEAR EQUATIONS 441 nature is known about nonlinear equations Our purpose in this chapter is to survey some of the central ideas and methods of this subject and also to demonstrate that it presents a wide variety of interesting and distinctive new phenomena that do not appear in the linear theory The reader will be surprised to nd that most of these phenomena can be treated quite easily without the aid of sophisticated mathematical machinery and in fact require little more than elementary differential equations and twodimensional vector algebra Why should one be interested in nonlinear differential equations The basic reason is that many physical systems and the equations that describe them are simply nonlinear from the outset The usual lineari zations are approximating devices that are partly confessions of defeat in the face of the original nonlinear problems and partly expressions of the practical view that half a loaf is better than none It should be added at once that there are many physical situations in which a linear approxima tion is valuable and adequate for most purposes This does not alter the fact that in many other situations linearization is unjusti ed2 It is quite easy to give simple examples of problems that are essentially nonlinear For instance if x is the angle of deviation of an undamped pendulum of length a whose bob has mass m then we saw in Section 5 that its equation of motion is 2 sinx 0 1 and if there is present a damping force proportional to the velocity of the bob then the equation becomes dzx cdx g 22539 O ESIDX 0 x In the usual linearization we replace sinx by x which is reasonable for small oscillations but amounts to a gross distortion when x is large An example of a different type can be found in the theory of the vacuum tube which leads to the important van der Pol equation dzx dx 2 t 3 dtz ux 1 dt x O zit has even been suggested by Einstein that since the basic equations of physics are nonlinear all of mathematical physics will have to be done over again If his crystal ball was clear on the day he said this the mathematics of the future will certainly be very different from that of the past and present p niIe1EeierrmL E1UeTION E It iwili be seen later that eeeh ef these nenlimear equetiene has inteireetinig iprieiperiLi eee met shared 0w the ethieire Threughseut this ehapteer we efhelii be eeneernied wkth eeeend enirder neniiinieer equaiiene of the ferm pv t u r r r p f Cs includes equetieins 11 2 and 0 es speeiei eases If we imagine a ieimpie dyeniemieeel system een5isiin ef aperIi1 e ef unit mess moving en the xaeEii5 end if fx k is the fm eie ieieti eg on it i hEI la Yf is the eiquetiieim eff metinni The mines ef J ip esiitini end elem Ey il l itiy lj P d em eeeh inetenti ehierieetieiriee the smite of the eysterni ere eaiied n and the plane of the 1r39 3I39ii iIiAES r and 39dr eeliedi the pheeee ptquoteeet If we intreiduee the ver39iiafbie y dfj tileen h can he repleeed by the 39Eq liiiV39EiE i eyeteim 5 We ehelil see P T ei geed deei een be ie ezmed 3 bquot iUiii the eeluitiene of l4i W etudyingi the eeiut39ienisi of UW when I is r39ege rder are a parameter then geiinierieai e seluymiien ef S is 3 pair of funetsierne t39if end yr de ning e euirve in the Jy plenei whieh is eifn1ply the phase plane memiien ed vli e quotWe I be imrereeted in the mteil piiemre formed by these ISL2L3lquotquotHquotE39S in the phase pleniei More gieiniereii we etudyr systems Def the feznni Fimi 6 Gixiyi where E end are eenmnueue and have eentieueue first iperitieii eleriiivetiveis Ihreiuhe uvt the plane A system ef this P in which the iI 1tCiEpE dEHt iveriainie I deee L l appear in the funetiens en i 1 en the riight is said In he eeIneewfus new turn I10 re ciieeer eeemimtiien ef the sieiiutienisi ef such a isyet39iem It E Uiie39wei fmrn em eeeurnptiiene and Tiheerem 54A that if t is any niumbef endi J I 3939g is any paint in the phase plane then ti39iere exiets e erniiq L1 eeiu trienn I E io ir 2 my Am NONLINEAR EQUATIONS 443 of 6 such that xt x and yt yo If xt and yt are not both constant functions then 7 de nes a curve in the phase plane called a path of the systems It is clear that if 7 is a solution of 6 then X xt c yy0d is also a solution for any constant c Thus each path is represented by many solutions which differ from one another only by a translation of the parameter Also it is quite easy to prove see Problem 2 that any path through the point xy must correspond to a solution of the form 8 It follows from this that at most one path passes through each point of the phase plane Furthermore the direction of increasing t along a given path is the same for all solutions representing the path A path is therefore a directed curve and in our gures we shall use arrows to indicate the direction in which the path is traced out as t increases The above remarks show that in general the paths of 6 cover the entire phase plane and do not intersect one another The only exceptions to this statement occur at points xy where both F and G vanish Fxm o 0 and Gxu u 0 amp These points are called critical points and at such a point the unique solution guaranteed by Theorem 54A is the constant solution x x and y yo A constant solution does not de ne a path and therefore no path goes through a critical point In our work we will always assume that each critical point xy is isolated in the sense that there exists a circle centered on xy that contains no other critical point In order to obtain a physical interpretation of critical points let us consider the special autonomous system 5 arising from the dynamical equation 4 In this case a critical point is a point x0 at which y 0 and f x0 0 that is it corresponds to a state of the particle s motion in which both the velocity dxdt and the acceleration dydt dzxdtz vanish This means that the particle is at rest with no force acting on it and is therefore in a state of equilibrium4 It is obvious that the states of equilibrium of a physical system are among its most important features and this accounts in part for our interest in critical points The general autonomous system 6 does not necessarily arise from any dynamical equation of the form 4 What sort of physical meaning can be attached to the paths and critical points in this case Here it is convenient to consider Fig 66 and the two dimensional vector eld 3 The terms trajectory and characteristic are used by some writers 4 For this reason some writers use the term equilibrium point instead of critical point mFFIaeteHTatL ElC39tMTtUN5 FIGURE GS de ned B Vh39EItt Ay trtatli T G39xiyi1 which act a tzyp ieI peittt P xJ gtI lJ39E yK1 hes htlFiEtZlnt eempette11t Ixe1t and tverttitee t2 mfp nEI39l t tG ry Sinee dt md dfdt 2 M this veeter M tangent te the path at P and points in the tdtirteetiett of ittsetleaeittg L If we th39i k 0f 1 as time then if can be int fpr t Ed ee the veteeity vetetetr net 3 particle mtevitn g etettg the path We can else trnagi1ne that the entire phase plane l letd wittth pertieitee E lJI1d that eaeht path is the trait ef a mating pettitz le PIEC t d and tetlewve by mtetny etihets en the same patht and eeeetmpenied by yet erthere en nearby pethe ettuetten eent he EhESvCl39Tib39E d as at twe tditmeneietneLl uid tm ett7en end eihee the s3ete m is tJ m I1tZJ E39l1 Li5 wihich meme that the vee1er V39xeyt at e tted perhtt Lye deee net ehamge with time the uid jrnetien tie tLtett397evrterjL The paths are the t1 ajBCI I iE5 of the moving particles and the etitiiee jp i 39tS Q w and S are peinte et zere veteeity ewhere the particles are at rest it e etagnetiert peittte eft the uid mtevtiteh Th meet etttilkimg tfeatutes of the iuitl 1 D139t itltusittettted in he ate PB the Critical peihte the erirangemert39t f the paths neat tirittieal petnttet eAe the etahiiitty er inetahititgs ef etitieel peittte tthet ts wzhethet a par39tictte nteet such a point tvetmeinet near er twendtetrte ei itite tetttether part ef the plane d ehtste d paths hithe in the t tguare which etetre5pert39i I periedie 3eluttetne NONLINEAR EQUATIONS 445 m all C nouns 67 These features constitute a major part of the phase portrait or overall picture of the paths of the system 6 Since in general nonlinear equations and systems cannot be solved explicitly the purpose of the qualitative theory discussed in this chapter is to discover as much as possible about the phase portrait directly from the functions F and G To gain some insight into the sort of information we might hope to obtain observe that if xt is a periodic solution of the dynamical equation 4 then its derivative yt dxdt is also periodic and the corresponding path of the system 5 is therefore closed Conversely if any path of 5 is closed then 4 has a periodic solution As a concrete example of the application of this idea we point out that the van der Pol equation which cannot be so1ved can nevertheless be shown to have a unique periodic solution if 1 gt O by showing that its equivalent autonomous system has a unique closed path PROBLEMS 1 Derive equation 2 by applying Newton s second law of motion to the bob of the pendulum 2 Let x0y0 be a point in the phase plane If xt yt and x2t y2t are solutions of 6 such that xt x0 y1t yo and x2t x0 y2t2 yo for suitable t and t2 show that there exists a constant c such that xt c x2t and yt c y2t EC39UA39T1 H5 P Describe that relatzian batwmena th jphasa pmtmits Elf the 5y5 tEmE 0 0q 5 The uritical points and paths of Eqgu ti U n 4 are by d4e nitim1 thoske 0f the equivalent sysrtem 5 Fin21 the critical pmints csIf Equati ns LL 3rit tl E3 Find the riri39taisaE pD ir E39I5 Hf and skttch in the x y Jp1ane same nf the cuuwres die inied 0 these 5Uiutinrn5 59 OF CRTlCAL P C mide an autnnanmitm5 sr5AtEm rFmm m dy A lg Gxy NONLINEAR EQUATIONS 447 We assume as usual that the functions F and G are continuous and have continuous rst partial derivatives throughout the xyplane The critical points of 1 can be found at least in principle by solving the simultaneous equations F xy O and Gxy 0 There are four simple types of critical points that occur quite frequently and our purpose in this section is to describe them in terms of the con gurations of nearby paths First however we need two de nitions Let xy0 be an isolated critical point of 1 If C xtyt is a path of 1 then we say that C approaches x0y0 as t gt 00 if limxt x0 and yt y05 2 gtoa Geometrically this means that if P x y is a point that traces out C in accordance with the equations x xt and y yt then P gt xy0 as t gt 00 If it is also true that j lt3gt exists or if the quotient in 3 becomes either positively or negatively in nite as t gt 00 then we say that C enters the critical point xy0 as t gt 00 The quotient in 3 is the slope of the line joining x0y0 and the point P with coordinates xt and yt so the additional requirement means that this line approaches a de nite direction as t gt 00 In the above de nitions we may also consider limits as t gt 00 It is clear that these properties are properties of the path C and do not depend on which solution is used to represent this path It is sometimes possible to nd explicit solutions of the system 1 and these solutions can then be used to determine the paths In most cases however to nd the paths it is necessary to eliminate t between the two equations of the system which yields QG dx Fxy 39 4 This rst order equation gives the slope of the tangent to the path of 1 that passes through the point x y provided that the functions F and G are not both zero at this point In this case of course the point is a critical point and no path passes through it The paths of 1 therefore coincide with the oneparameter family of integral curves of 4 and this 5 It can be proved that if 2 is true for some solution xt yt then x0 yo is necessarily a critical point See F G Tricomi Differential Equations p 47 Blackie Glasgow 1961 D39FFEREHTML tECUttA T39U39ihl5 family man ahftah he whtaiLnad Y the mettLhad5 hf Chtaptaer It ahntuldt nh ted h wevaar that white the paths f 1 are ditrtactad cuswaa the integral cuwea at 4 have no dtiracttiant taasactattad with them Each of these taahh iqt1Eat far datarmitningj the paths wail ha itlluatrxatad in the aaamiea below haw gitva ga mtatri c ctasartpttan5 at the tan math ttypaa at Eritiaal pauintaia In each case we aastume that the mritticat paint urznd etr diszzuaainwn ma tha Origin 0 r t Nudels p vstrhittzaat paiattt like that in 6 is ctalhadt a adE Such a hint 6 appmachad7 and alan anttatmd 4 each path as I4it wt or as I 3 Far thata made 5hnw39nt Fig a639E39 than ara faur halfline patthsz p B0 K and DD twhich tta gtathar with the avriairt make up the lines A and 0 At mhar paths reaamta parts of ptarahata5 hand as each of these paths ttp JTUHJ hE5 0 its staph tatpmatchaa that 0f the line Example CDnaidar tha system 0u It is atestar that the arigixa is the minty E3I39iEi E VW paitzrttt and tha gEnarat aaluthan can he fauhdl quite taa5ihr hy the tmethhdis f Sactihn 56 JE p 7a k I b 3 aa quotA tc1a339t when cq t we htava gr U and y 3631 In this case tha path Fig 63 P P thua pa5quoti39thra yaaia wfhan Ac 3 at and tha ntaga tiva y aaia quotw han t g quotif a and aaah patTh tatppraacthaa and enters the origin as t am Wham C I we htawe 2 me and y mathquot This path ia the haIHinE y tquot tr E 11 whant an 3 0 and the hta elgitlta 39 2 1 0 when cg and again hath paths agppmaEh and enter tha vcrrigin as I 9 an quotWhen hath 6 p E3 an 0 the paths he an tha paIrathAaiaa y I Eyquotszfjlalg which gt thmugh the arigin with alhp a it It sihhhtad ha utndaratata d that aach of tthaaa patha anastists of manly part if a patrtahma that part with at 3 D if at 33 o arl the part 39writh 1 4 PJ if 274 a H Each at 39tthaaa paths atlao apptthtracthaa and antara the 39fig E as t in am thja can ha shah at ah1 frcrm hIt If we p39l39tJC39Ef 1d diract1y fmm 5 In fliha di arantia aqLtatttian at 1 T NONLINEAR EQUATIONS 449 all FIGURE 68 giving the slope of the tangent to the path through xy provided x y 00 then on solving 7 as a homogeneous equation we nd that y x cxz This procedure yields the curves on which the paths lie except those on the y axis but gives no information about the manner in which the paths are traced out It is clear from this discussion that the critical point 00 of the system 5 is a node Saddle points A critical point like that in Fig 69 is called a saddle point It is approached and entered by two halfline paths A0 and B0 as t gt 00 and these two paths lie on a line AB It is also approached and entered by two halfline paths CO and D0 at t gt 00 and these two paths lie on another line CD Between the four halfline paths there are four regions and each contains a family of paths resembling hyperbolas These paths do not approach 0 as t gt 00 or as t gt 00 but instead are asymptotic to one or another of the halfline paths as t gt co and as tgt00 Centers A center sometimes called a vortex is a critical point that is surrounded by a family of closed paths It is not approached by any path ast gt00orast gt 00 Example 2 The system 8 Elliquot1EHEHT39l 39LE39UAT39IUHi5 FGiLTRE 69 I135 the migin as Ms nly critiuzfa point and its general 39uluAtim is I an sins czcmsaf W V 2 cc1s1r 1 sim The sD mian satisfying the rmandE tiun5 139I2ls 1 l d Ly jw E E is clearlyr Jr 2 cfasr DJ J 2 Sim j and the 50iu39Ei n LiEFEEI39E39 39i3939iEd by 1r39lfCl39 Y U and u 11 is z J1quot 11 am I rsus I E7 y quotE 39i395i39 2 ma r P Th two d7ifferent suiutiuns dir nE the amE pam IE 39Fig TH which 175 Evide nt y the uzircle 139 y Bn1h IQ and I1 Shaw tfmaft th39is paI h is traced wt in 1h 39 uunterc Emckwi13 H39rtiun If we ielliminate E 39E39EWEEr39l the Eq39uatjmnV5 Bf thc 51ste4m we gait A as p 0 whuse gvenem l L1 wtinnAx2 Ly 43 3rieIid5 all the pai39hs buVt WiIquot139NIE their IquotEEMUvHS It i5 ampjhvE LiE mat the uri tinai jprJinAt LII J39 M the system 8 is an EEJ391lquotZE NONLINEAR EQUATIONS 451 FIGURE 70 Spirals A critical point like that in Fig 71 is called a spiral or sometimes a focus Such a point is approached in a spirallike manner by a family of paths that wind around it an in nite number of times as t gt 00 or as t gt 00 Note particularly that while the paths approach 0 they do not enter it That is a point P moving along such a path approaches 0 as t 9 00 or as t gt 00 but the line OP does not approach any de nite direction Example 3 If a is an arbitrary constant then the system d lt12 Q xa dt y has the origin as its only critical point why The differential equation of the paths dy x ay d x ax y 13 is most easily solved by introducing polar coordinates r and 9 de ned by x rcos 8 and y rsin 6 Since r2 x2 yz and 8 tan 39X x 0 N UlFFEREHT IhL EaumT1u3Hs TI we SEE Ihm dr f I 6 0 Q P it it Wim the aid f Ithezsve EquV3tic1njs 13 can easily The wrjitten in the very 5 impIse farm dr E Hr 9 5393 V r 63quotquot i5 ugh P913 gquaitfi fl of the p1 th5 The two pE5a5 iMIE spiral L ni g uratinn5 am Eh wn in Fig E and the clirectia1n in whiczgh Ihescc path arc traverse d can be 586111 fr m the tax2 t hat g 0 F Vy whpesn 1 A D If m P 9 lhen UZJIA ctj ames tn 8 and 14 henme5 1 which is the p rlar eqluatiuran nf tlm family xi y E If az lv Ein 5 c39nmredI on the c1r39igirm This example thtreIfare genemjliz e5 E3ample 2 and Since the centcr Shawn in Fig Tf stands DH thc 39390rdLerwIine heEwe en the 5piraEs f F ig B 3 cristirzza pmint that E5 3 ienter is ften Called a bmdEr fne 515quot We will Imrm1ntampr nthm39 bordserl39ine cases in the max s1ectinn We rn w intr dAucE ha mn epl of smH39ry as it Aappiies tn P critical pDiJi391tS of ma 53rsmm 1 was35 painted mum in the qprmrimzs sEctinrm that one of wthe mast imjp rtantlz q uesmiums in the study of a Lphysi c al szystem is tAhaIt waif M5 steady smte5 Hgwgugn 3 51gadAy state r littl7 Aphyws zai 5ig11 V aame mraljaass 0 has u m39ga5 nabIE idgggeg n f39 grmanetmce E unless it is statl c As a simple NONLINEAR EQUATIONS 453 YIl t at X f K TCJ agtO altO Q all FIGURE 72 example consider the pendulum of Fig 73 There are two steady states possible here when the bob is at rest at the highest point and when the bob is at rest at the lowest point The rst state is clearly unstable and the second is stable We now recall that a steady state of a simple physical system corresponds to an equilibrium point or critical point in the phase plane These considerations suggest in a general way that a small disturbance at an unstable equilibrium point leads to a larger and larger departure from this point while the opposite is true at a stable equilibrium point e We now formulate these intuitive ideas in a more precise way Consider an isolated critical point of the system 1 and assume for the sake of convenience that this point is located at the origin 0 00 of the phase plane This critical point is said to be stable if for each positive number R there exists a positive number r S R such that every path which is inside the circle x2 y2 r2 for some t to remains inside the anaajiau ucu1 mt FIGURE 73 454 DlFFEFtE sTleL E ZHLaTlvDiHS FIGURE 1394 eirtele xi i y R2 fr all I 2 is Fig M Leesely spealtinlg a eritieal peirit is stable if all paths that get sruf eiehtly elese te the paint stay elese tea the peirll F39urthe139 taut eritiea I paint is said the he iIsympI tie el39iy stabie if it is stable and there exists a circle I2 y r stieh tthat eirer r path which is inside this ritele fat some I ti appreaehes the erigin as t 2 W Finally if eur etitieal paint is 11eit stahle then it is eallecl urisiabie As examples eit these eeneepts we peint eat that the nede in Fig 6 the saitldle point in q 69 and the spital en the left in Fig 72 ate iinstahle while the eenter in Fig quotill is stable but met asjymptetieaIlyi stable The neele in Fig the spiral in Fig 7i and the spiral en the right in p2 72 are ESt mp 39it39aliy stable 1 1 Fer each rat the fellsewing netilinear systenis l find the etitieal paints ii lied the di eriential EEl1E1ii I7l ei the lpaths iii some this equatien to find the paths and 0 sketch a few ef the paths anii shew the tiirei tien elf illEtquotE SliIllg t N I 2 Each ef the felleiwjing linear systems has the erigin as an iselatentl eriitieal peint ii Find the general selutien ii Find the ditferentia eqiiatie n elf the paths NONLINEAR EQUATIONS 455 iii Solve the equation found in ii and sketch a few of the paths showing the direction of increasing t iv Discuss the stability of the critical point 525 E5 dx 4 dt quot dt dt y 3 b C y 1zy zx dt dt dz 39 3 Sketch the phase portrait of the equation dzxdtz 2x and show that it has an unstable isolated critical point at the origin 60 CRITICAL POINTS AND STABILITY FOR LINEAR SYSTEMS Our goal in this chapter is to learn as much as we can about nonlinear differential equations by studying the phase portraits of nonlinear autonomous systems of the form dx F dt x y dy G dt x y One aspect of this is the problem of classifying the critical points of such a system with respect to their nature and stability It will be seen in Section 62 that under suitable conditions this problem can be solved for a given nonlinear system by studying a related linear system We therefore devote this section to a complete analysis of the critical points of linear autonomous systems We consider the system dx ampquott a1x 1 53 b gdt a2x 2y which has the origin 00 as an obvious critical point We assume throughout this section that 11 b1 0 2 a2 b2 so that 00 is the only critical point It was proved in Section 56 that 1 has a nontrivial solution of the form X Aemf y Bemt 455 sIFFEliRElHTmL Ettulanlnws twhlenervcr quotis a rut oft th iquatlrtatlic cqluatimt quot 0 0 p 0y quot39 u quot739 Uh pn whithe is ll llil that tIll5H F egqumiant of the zsystcmtt ObElElTVE that t39ntlititl39n n implies that ziem canttmt he a mat f 3t lget hm and ma that the rmzitsl at 3 W6 shall golrmla that th nature inf thee critical point 10 of thc systam L1 is determined by the nattttc of the fnuimers ml and H125 It is F3350nlahlle to expect that three puJrssailJilit i s will occur accolrding as mi and m3 are real axnclt ldlistinczt real and equal or mnjulgatte complex Untfmtmiately the situatiunt is 3 little mute mmplicated than this an it is necessary to mnsitdelri ve cases subdivided as f nll0wsl Majar rages Cate E The mots mnl Hindi m3 are real diminct and f the same Sign T hadB T Calm p The Irtmts ml and lrnz are real di5tinct and of 39fpquot339USlltEA stigma saddle point Cute 0g Thc rmtls ml and m2 are mnjiugate mmplex but mt putt iniagilnary lspiralt Br39 lquotquotdEli39I iI39II 39f cases CREE 0 The malts rm and N12 are real and Equal nluJde Case The rants mm and are pun imaginary center Th raasnn f cr Ih di3tintti n httween lh tmajlvar cases and the hwzrrtlartlitnc cases will ettnme clear in Sectian p k For the pittataint it suffices to mmalrk that while the hb UFdEl39Iil cases are at mEt39tl1 m tl l tintelrest they have little 5lgI391Ji t3 1 fur appliciatlilons b ll the cir ct1lmlstan ces tle nit1g ttham am L1nllikaly ta Blfl in physficail pmbllem We naw turn tn the p139mits of the assertions in plEiI Ei1Iil391eS ES Ea5e If the rlmits rm ancl ml an real distinct and of tlie same lagn than the critiacal point 00 is 21 mnde Prtmf Wt ihcgin hwy 3s5t1ml ing that rm and int arr bath t1 gti1Zlquot39rquotE altd wt Eh the tltItt llt1rt39i 50 that rm 4 mg b LL By SEWlEitiJII l Z the getteml snlutiane at 1 in this cats is I c4E quot1quot39i E c2A3E quot3 4 y lcB quot quot E39 1H3Em3i wllet the A39s and ti 5 m rjltz nitttt tjtzmattints 51lL1l I that TBHIAE B3I A and where the l 5 are arh itrar 3r mnstanlts when 2 L we Ubil l the S lt1llt tl5 I cA1 ei quot39 Z El ErIlFr 5 Azy 457 NONLINEAR EQUATIONS and when c O we obtain the solutions quot y T39 For any c gt 0 the solution 5 represents a path consisting of half of the line Ay Bx with slope BA and for any c lt 0 it represents a path consisting of the other half of this line the half on the other side of the origin Since m lt 0 both of these halfline paths approach 00 as t gt 00 and since yx BA both enter 00 with slope BA Fig 75 In exactly the same way the solutions 6 represent two halfline paths lying on the line A2 y Bzx with slope B2A2 These two paths also approach 00 as t gt 00 and enter it with slope B2A2 If c O and C2 O the general solution 4 represents curved paths Since m lt 0 and m2 lt 0 these paths also approach 00 as I gt 00 Furthermore since m mz lt 0 and cA2equot c2B2e quot239 6 y cBe quot39 c2B2equot392 cBc2equotquot quot2quot B2 x cAequotquot39 c2A2e 392 cAc2equotquotquot 392quot A2 it is clear that yx gt B2A2 as t gt 00 so all of these paths enter 00 with slope B2A2 Figure 75 presents a qualitative picture of the situation It is evident that our critical point is a node and that it is asymptotically stable If m and m2 are both positive and if we choose the notation so that m gt m2 gt 0 then the situation is exactly the same except that all the paths now approach and enter 00 as t gt 00 The picture of the paths Yll A yB1X 31 FIGURE 75 4518 D WEREFquotlAL EtD3 L3tTtiJH395 giveen the Fig T5 is unchanged exeept t het the etrreers ehewing their dttteetinnet are all retrereed We etiti ha t39E e eerie but nw it is unete blet Caste B It the tents rs1 end F are reeL dti5tine ttt and et e ppeeite Signs thena the aeritttieet pDi3939tlZ gtll is a sadidle tpe intt tPreeJ39 We rmeyr Cheese the netetiieh ee that mm ii Ur if tm The general eetutien ef it can still be written in the term 34 entI again we have p tquott39ittlIatquot eettt tiene ef the ferf39rne 5 et39Id E The twee ha39lJfTline paths t ept e5ente d hjf 5 still etppretaeh and enter UtD as I I 09 but this time the twee htettttvttitne paths rE p r39E i d Iiby 6 eppreeeh and enter VIEttI17I ee t P m eg at U and E1 4 E1 the generate eeltuttent 4 still r39epreeenzts eutrved paths but since Amt e H vi mg nene ef these petite eppt e eehets t U x 1 at er I 0 Itteteed as t 1 tie eeeh ef thee jpethhts is aem pttetie te meet ef the halfi39ne 3I39Hf l1E repr39eset1teat by 6 and as c Em each is a5ym39tettte te ette ef the ha itizne paths 39tIepreeented by 5 Figure F6 ght S at q uteltitettiitre pitetute et k b39Ethv 39t39iUI39 In this eeee the eritieelt peint is e 5et1tiIe point end it is eh1vquotitetu5ty u tItstt1htet Ceee It the tenets ti anti em ere teetIfj uAgette eernplet hut tnaet pme imagitneryt then the etijtieet p39tEiI t L IEl its a spirre t Pu In quotthis ease we can hwrite m and mg in the fetm e i the where at end l are t39IUl39tEEIquotU reel ntumbere A13 39E fer later use we Dhs that the Fl39l539U WE NONLINEAR EQUATIONS 459 discriminant D of equation 3 is negative D as b22 4alb2 025 C1 b22 402 lt p By Section 56 the general solution of 1 in this case is e 39cA cos bt A2 sin bt c2A sin bt A2 cos bt x y equot39cB cosbt B2 sin bt c2B sin bi B2 cos bt 8 where the A s and B s are de nite constants and the c s are arbitrary constants Let us first assume that a lt 0 Then it is clear from formulas 8 that x 9 0 and y gt 0 as t gt 00 so all the paths approach 00 as t gt 00 We now prove that the paths do not enter the point 00 as t gt 00 but instead wind around it in a spirallike manner To accomplish this we introduce the polar cordinate Band show that along any path d6dt is either positive for all t or negative for all t We begin with the fact that 9 tanquot39 yx so i xdydt ydxdt dz x y2 and by using equations 1 we obtain Q9 azxz b2 wary by2 dt x2 yz 39 9 Since we are interested only in solutions that represent paths we assume that x2 yz 0 Now 7 implies that a2 and b have opposite signs We consider the case in which a2 gt 0 and b lt 0 When y 0 9 yields d6dt a2 gt 0 Ify i 0 d6dt cannot be 0 for if it were then 9 would imply that a2x2 b2 axy by 39 0 or 2 a25 b2 a5 b 0 10 y y for some real number xy and this cannot be true because the dis criminant of the quadratic equation 10 is D which is negative by 7 This shows that d6dt is always positive when a2 gt 0 and in the same way we see that it is always negative when a2 lt 0 Since by 8 x and y change sign in nitely often as t gt 00 all paths must spiral in to the origin coun terclockwise or clockwise according as a2 gt 0 or a2 lt 0 The critical point in this case is therefore a spiral and it is asymptotically stable If a gt 0 the situation is the same except that the paths approach 00 as t gt 00 and the critical point is unstable Figure 72 illustrates the arrangement of the paths when a2 gt 0 Case D If the roots m and m2 are real and equal then the critical point 00 is a node X iD irFFE1EEsT IAL E IU T i 1 39TS Ml Pi39 39f We egin by sssumin stehst mi ems s m s I There are two subeasess that require sepmites diseussien 5 an 2 ha 9 Ii and e 5 U ii all DI 1quotEeI pIJssi39biities lessding in a dmlhie rem eff I 3eCU 139ITiI3M39I p We rsrt eesnssiders the su bease ti whixeh is the sitlsetien deseribezd in the feetn ete in SEEtiD39 56 If e denetes the eemmcm value ef a seed bi men E 13 Ei IquotI 3 beenmes ml e a3 0 end m S e The system 1 is Elms H 0 and its general se39lmi n J E eequotquot V A Po W L I 9 H where rEA am sea ere srhistrsery eensIfants The pe1zThs de need by 11 are half hijtnses ezl all pessiblse rslepes F igs 39F Md since s II we see that eeeih path F39PET Ch39E5 and enters IIJ7 es re 26 ms The kelrisnieal pain is mmefere s nsedes endl it is a5s quotmpt 39fiE Nf sIsJ eIe If m 2 11 we have the same situmien eseept that the paths entesr avJU es E P ss the arrews in Fig T are resversseds and I1 is tmsiieble FlG IJHE TI NONLINEAR EQUATIONS 461 We now discuss subcase ii By formulas 5620 and Problem 564 the general solution of 1 can be written in the form x cAequotquot c2A Atequotquot y cBequotquot c2B Btequotquot 12 where the A s and B s are de nite constants and the c s are arbitrary constants When c2 0 we obtain the solutions 1 c Aequotquot 13 y cBe quot We know that these solutions represent two halfline paths lying on the line Ay Bx with slope BA and since m lt 0 both paths approach 00 as t 60 Fig 78 Also since y x B A both paths enter 00 with slope BA If c2 O the solutions 12 represent curved paths and since m lt 0 it is clear from 12 that these paths approach 00 as t gt 00 Furthermore it follows from X cBe39quot c2B Btequotquot C18C2 B B x cAe quot c2A Ate cAc2 A At that yx gt BA as t gt 00 so these curved paths all enter 00 with slope BA We also observe that yx gt BA as t gt 00 Figure 78 gives a qualitative picture of the arrangement of these paths It is clear that 00 is a node that is asymptotically stable If m gt 0 the situation is unchanged lt AyBX FIGURE 78 NONLINEAR EQUATIONS 463 If we now write equation 3 in the form m m1m m2 m2 pm q O 15 so that p m1 m2 and q mlmz then our ve cases can be described just as readily in terms of the coef cients p and q as in terms of the roots ml and m2 In fact if we interpret these cases in the pqplane then we arrive at a striking diagram Fig 80 that displays at a glance the nature and stability properties of the critical point 00 The rst thing to notice is that the paxis q 0 is excluded since by condition 2 we know that mlmz ab 0 In the light of what we have learned about our ve cases all of the information contained in the diagram follows directly from the fact that p rt Vpz 4q 2 mumz Thus above the parabola p2 4q 0 we have p2 4q lt 0 so m and m2 are conjugate complex numbers that are pure imaginary if and only if p 0 these are Cases C and E comprising the spirals and centers Below the paxis we have q lt O which means that m and quot12 are real distinct and have opposite signs this yields the saddle points of Case B And nally the zone between these two regions including the parabola but excluding the paxis is characterized by the relations p2 4q 2 O and q gt 0 so ml and quot12 are real and of the sameisign here we have the nodes of Cases A and D Furthermore it is clear that there is precisely ASYMPTOTICALLY UNSTABLE STABLE STABLE Spirals Centers Nodes FIGURE 80 C D1EFERENma1EtjuAn0H5 DEB regim1 nf asympt ti c stalaiIAitr tbs mt quadArant WE state this fm ma ly 35 fEJM WS Th mrrem The cririmf p i39nI nf HIE HrIE 1r39 sysrem 11 3 tI539mpf E fE Hy mb39E f n d n v lfIJFlE u E i39En39 3 p aria hi and 3 A lbg ugh Hf nquot i E mifiary equ mfmi p care bath pa rite FinaIl3 39 it shnAuld be emphasized that we have Studied the paths f mu lincar system near a criticaE int by Vanal3u mg exp39licit solutinns Elf the sjrVstuEm In the nmct twc Wcti ns we enter mare fully into the spirit of the subjacut Alzny inJmsVtigatimg 5imiliagr pas bleVm5 fur nnnfinear systems in gen3IaI cannmt he 5 nlvead explicitiyV Dit39em1ine th na39turE and setabilinr pmpuertzies Uf th cri39II39ital pninrt 00 for E h of the f Nmwing Iiinuear aummmu1ru5 systems l if b 4 4 S at r B 5r2y 1 dz 0 J E l1r 5 A La E if 4 g f d E39 492 F E Edquot I J V 4 P Li39 ErF39 J5 I In H II P 02 PX 2 If igb 2 nib 2 pT show that vzhe 5rstVem MhasiAn ni1eir many critilcai pmums none af which are imllaiedt 3 If b E325 p U ahnw that the jrquot5E39rBTFl gr I J I39VE brquot an x gy 1quot has a single isolasted c139itiVEai paint xmy NONLINEAR EQUATIONS 465 b Show that the system in a can be written in the form of 1 by means of the change of variables 2 x x and 7 y yo c Find the critical point of the system dx 2x 2 10 dt y ix d 11 8 49 dt y write the system in the form of 1 by changing the variables and determine the nature and stability properties of the critical point 4 In Section 20 we studied the free vibrations of a mass attached to a spring by solving the equation dzx cbc H22bEa2x0 where b 2 0 and a gt 0 are constants representing the viscosity of the medium and the stiffness of the spring respectively Consider the equivalent auton omous system dx E dy E a2x 2by which has 00 as its only critical point a Find the auxiliary equation of What are p and q b For each of the following four cases describe the nature and stability properties of the critical point and give a brief physical interpretation of the corresponding motion of the mass i b 0 iii b a ii0ltblta ivbgta 5 Solve equation 14 under the hypotheses of Case E and show that the result is a oneparameter family of ellipses surrounding the origin Hint Recall that if Ax Bxy Cyz D is the equation of a real curve then the curve is an ellipse if and only if the discriminant B2 4AC is negative i 61 STABILITY BY LIAPUNOV S DIRECT METHOD It is intuitively clear that if the total energy of a physical system has a local minimum at a certain equilibrium point then that point is stable This idea was generalized by Liapunovb into a simple but powerful 6Alexander Mikhailovich Liapunov 1857 19l8 was a Russian mathematician and mecha nical engineer He had the very rare merit of producing a doctoral dissertation of lasting value This classic work was originally published in 1892 in Russian but is now available in an English translation Stability of Motion Academic Press New York 1966 Liapunov died by violence in Odessa which cannot be considered a surprising fate for a middleclass intellectual in the chaotic aftermath of the Russian Revolution 0 D FFEEE39HTiAL ED39U39Aquott t HiS T for stutdyintg stabtIitjy pfDblEm5 in a bf d f tDttIe xt WE shall diSt ttSS Liagpuntntr 5 mtetttlmd attr l Same mt its apgplicattimrts in this and the tnextt siectiant ansid ter an aH tK l m u15 39sy5tEmt Fi MN t w M and atsume tthatt this system ms an isolated t li l tpt itnti which as ttsuatlt we EEIIEE to be the Driiirm UtD539 K K 2 rt be a path EFif39l1 and C StdE t a futnmuzm Eritt thatt is cnnstintmm and that Etjintit tllilttt first partial d tfitn39ELttv39EE in EL regitunt mtntainittg thlli p Hth If a paint txyr mtmes taking ttt patttht itn attmrdanrsE with the eqt1ait imrts 2 xt and y 4B tthtert IfJy can be r egar dE d as 31 tfutncti nt B t jIt1Iquot1g C WE detnte ifunmimt 0 E t5t l39I d its rate of change is W Emmtm t t t U F G This f fmm is at Ih heart of Liapunmrs ideas and in t 39tdEI I113 tmptlnitt it we tteezd 5Eta39etat1 de nitinnst that zstpctvctttfy Ih kinds at functtimts we shall be i TEEtSTECI 9 Stp pasc that y 1 trIJIrltifiquotlIt quotLiI5 anti has CU Ei tUrE39UtS p 5 patttiatl detrivatitvEs in 50im tragtimt tcuntattinitntg tit uriginz If E traantisthses at the migin star that ET I l U Ehv it is sat tttm he D tLIt7 tJ39E de riftte if Etxy 1 J for txty it 00 and nggattvt de tztim if E139y4 at It fart xy DU Stimittartim 0 it catttetd gjttmitiw 3em d e m te if E U GI and 0 0 far 1y Dt and negtatiue tttemi tte rtittet it E1JU 0 and Etxs 1 far try 0130 It 7 Clear that futncti ns 0f the form mtg 4 byb twhare H and b are p isiftiiwi mnstantts atndi m and H are pnsititve ii IEg t39S are tnsiitive de t1ittampt Sitttctt E xjy7J is I7IJ39Eig3i i39E d tntitte t and mnly p ExLy i5 p sittive dE ttnLEt et functimfns tjrff the farm tax byyh with ta v D and b c U BITE ntegattiw d E tlmiltl n The tIttnEtiCItntS txE t 35 Zlt d I a yfm are that pusi ttv e dte tttittat but am ntssvtartthetless tttstttive 539 f 39Tilit tt pmntt 4rn 3r can atlwtays be nitwed tsu that t1I rigit39I by 3 tstmpt lr ansa litn Ctf tnjt39wrdjrta39t 3 2 2 1 ti ar1dF39 3 MM 50 tiherc is mm 1055 ttf gnurati t y in a55ttming that it HES at HE mrtgin in that mitt planet NONLINEAR EQUATIONS 467 semide nite If Exy is positive de nite then 2 Exy can be interpreted as the equation of a surface Fig 81 that resembles a paraboloid opening upward and tangent to the xyplane at the origin A positive de nite function E x y with the property that 8E 3E F G 3 8x 8y is negative semide nite is called a Liapunov function for the system 1 By formula 2 the requirement that 3 be negative semide nite means that dEdt 5 0 and therefore E is nonincreasing along the paths of 1 near the origin These functions generalize the concept of the total energy of a physical system Their relevance for stability problems is made clear in the following theorem which is Liapunov s basic discovery Theorem A If there exists a Liapunov function E x y for the system 1 then the critical point 00 is stable Furthermore if this function has the additional property that the function 3 is negative de nite then the critical point 00 is asymptotically stable Proof Let C be a circle of radius R gt 0 centered on the origin Fig 82 and assume also that C is small enough to lie entirely in the domain of de nition of the function E Since Exy is continuous and positive zit X FIGURE 81 DF39FERENT tL E UhTIJH 5 X it Iquot39lGLliLE 32 de ttitett it has re pesitive minirnurrn he en Eur Nett is erentinuens at the erigin and vanishes there se we earn nd est pesritive nnrnher r We R such that Ery lt m whenever x is itnrside the eirele C3 et rati1ius rt New liet B rtt yIJ be any path whieh is inside C3 fer z in Then Em a re and sinee is netgetiive sernide nite we have riE d D which implies that Eft E39ti cf fe139 elil I 22 Eh it feiiews that the path It can tntetrer39 reach the eir ete fer einy ti 3 I se we have stehility39 Te pteve the seeetnd pert eat the theer39ern it su iees te shew that tender the edditiensI 1i1jfp 391IhES iiE we eise have EU r 1 fer sinee Arty is presitiwe e1ef39initre this wilt irnpiy that the path C appreeehers the eritieel peint We begin by ehserving that since ti fdt rs I it fellews that Eff is a decreasing fumetien and sine by hypetrhesis E t is hnunded helew by 0 we eenseiude that tEt appreeehes setne iitnit L E 0 es t 2 mt wewe stthst E 4 I it su iees te shew that L A se we assume that L El and deduee a eeirrttreriieiti ent Cheese at pesitive number F lt r with the prepestry that Ery 1 L Whenever I is inside the Circle with ilreditrs F Since the funetien 3 is eerntinuees and negative de nitet it has at tnegaitritre mssziirnttnt l r in the ring eensistitrg ef the eitteles E end C3 end the regien hetwreen there This ring eenteins the E tiif path C fete t E rm she the equetien Em Etnr J e yiefds the inteq uaii39t3t H EU p Eire W tn 4 fer all yr 2 tn Hewesretr the right side ef 4 beeernes negartirveiry in nite es NONLINEAR EQUATIONS 469 t gt 00 so Et gt oo as t gt 00 This contradicts the fact that E x y 2 0 so we conclude that L O and the proof is complete Example 1 Consider the equation of motion of a mass m attached to a spring m aaao 5 dtz Cd 39 Here c 2 0 is a constant representing the viscosity of the medium through which the mass moves and k gt 0 is the spring constant The autonomous system equivalent to 5 is dx 2 dy k c 6 Zquot39 quot and its only critical point is 00 The kinetic energy of the mass is myz2 and the potential energy or the energy stored in the spring is jkxdx 1kx2 0 2 Thus the total energy of the systemquot is 1 1 Exy imyz Ekxz 7 It is easy to see that 7 is positive de nite and since I quot11 I Q I k c kxy my x y m m cy2S0 7 is a Liapunov function for 6 and the critical point 00 is stable We know from Problem 604 that when c gt 0 this critical point is asymptot ically stable but the particular Liapunov function discussed here is not capable of detecting this fact 8 It is known that both stability and asymptotic stability can always be detected by suitable Liapunov functions but knowing in principle that such a function exists is a very different matter from actually nding one For references on this point see L Cesari Asymptotic Behavior and Stability Problems in Ordinary Differential Equations p 111 Academic Press New York 1963 or G Sansone and R Conti NonLinear Differential Equations p 481 Macmillan New York 1964 rm n1rFEHEHT1 aLErumUH5 EmmpIE The System 3 has 00 B an iT5Dlated critical p i nt Let us try 0 prmE 5ta bili1ty by ElUHS Til39HC Ei g a Lirapumv functicm Uf me farm Equotxy ia7rEquot by It is altar that 135 b 5 Ex F P G 2rr339axl 39quot v2ryA nby1 quot r393 E y3 54m Iquotquoty 1 2n5bry quot Entbvyh we wish tn make the expressiun in parentLh e5es 1aanish and irm5pe1Ii u m shnws hat this can Ezra dame by chmsing o 1 n L 1 E L and K pY with theTse chnices we haw E xy E 2 E 3 which pDsitive d7e r1it e and r 3G VE4y which is negatiw 5E39midamp nilE The uritical pom D39 0f the syrstern 82 is thersf rt sIa39l1Ee It is clear fmm lhi Exanlple 1haVt in cnmplicaI EdJ 5i39T39LlatiIiII391S it may39 E van i 7 icul1I inde ed39 In mnstruct Euitabie Liapurmv zfuntinn5i The fljllmwing mum s rruetimas helpful in this connectimn T39henrEm E The fun rnnn Exy y 1u39y cy is p7mime d niEe39 9 and wily if a vA D and bi 4 E39 U and is negm iVve de nEre if an nnfy if E 11 and b3 435 395 EL Pw i f If y CI we have EIf lCl ME 50 ErUI 2 U f r 1 4 B if and uni if a i39 0 If 3 N 0 wer Lhaw y3 39JEE n WC CL and whE n 1 2 1 the bI EkEIEd polynn1mia1 in y whim 0 p sitivve fur large y is punsitive far all xfy if iaa d t 1 I1lIquot if if 4m I5 5B This pTD39tquot39BS the r539I part fquot the theorem and the secrcmxl part fu nzws at once by IcwI15ideri11g the fLl139mC39tiD Esy A y DeLirmim1E WlquotiE39Ih ET each of the Fen Dwin g 1139l39fiD n5 is pDsitim de ni1e 1Egatiw rie ni te EH neiiherz Z 0 J E r 35 r rm 23 1 4 M my x E axy 5y P Shnw theat a functiT31 f the farm cl bxzy cxf arty cannnt AltE Eith f pmrsitive dc nitc Ur nErgati vc de nite NONLINEAR EQUATIONS 471 3 Show that 00 is an asymptotically stable critical point for each of the following systems dx 3 I dx 3 21I3x y E 2xxy 8 b EX X5 2 3 CiX x2 2 3 it y dt y y 4 Prove that the critical point 00 of the system 1 is unstable if there exists a function Ex y with the following properties a Ex y is continuous and has continuous first partial derivatives in some region containing the origin 19 E0 0 0 c every circle centered on 00 contains at least one point where E x y is positive 39 d BE8xF 8E8yG is positive de nite 5 Show that 00 is an unstable critical point for the system dx 2x x dt y d x y5 6 Assume that fx is a function such that f0 O and xfx gt 0 for x 95 O that is fx gt 0 when x gt O and fx lt 0 when x lt 0 a Show that Em gr ffltxgtdx is positive de nite b Show that the equation d 2x 5 air has x O y dxdt 0 as a stable critical point c If gx 2 O in some neighborhood of the origin show that the equation dzx 1 gltx jiffltxgto has x 0 y dxdt 0 as a stable critical point foo 62 SIMPLE CRITICAL POINTS OF NONLINEAR SYSTEMS Consider an autonomous system FmH dt d U lGm dt 472 D I FFE EJ IT39EAL E 39U AT IUHE swish an isestsed eritisea pint st lIUr If siy and aGxe me be esssndsed is jpewser series in x ands y then lj takes the form E V I 13131 bi 04 e39g12 e dgxy Ely Z K s and M are samsellkgthsset is when is close to the migin the terms ef second degree sr1 higher are vsesrye smalls It is thereferse nature In diseased these nelnlinessre stermss sand Ji EjtquotM1 FxE39 that fthes rqeus tsstisvee ehsvim39 at the paths ef 2 nsesaar the eri39tieal paint ssirmeislsre lie that of the paths of the flquotEl t li limes syst em M Wes shelf see that in generel tshis is seeteusH1r the ease The preeess exf replacing I2 byr the hisnsesif s39sstem 3 is sussuslly called h neeeri ess J39en MD39TE genersl1y we shall eensider systems ef the farm es aMbwfMN EswbwsmMs W 4 will bee ssssumesd that HI bquot m9 L 5 3 3 H se that the reilsrlzsed linear ssystem 3 has 10 as an iselated eritieal paint g and gsy are E3 UIIquotiH39UDU5 sndi shave eenteiinueeuss rst epsmssls desriese39tives for all sys sud lr39hst as x s m ja we Zhsse gir MPf Observe ernditisens 6 ismply that fDs 1 sesnd gUU J se BM is a erities peint 01 4 slse it is nt dii ieults is wise thsst Ithis erixtieisi p39Oi ll is ise sted see Pe1eblemI1 1 the ressetreietirems listed i I sense U as sssid to be as sirraapie esrifyrielsf puss of thee sgrssterrns 4 ill an d lim 6A NONLINEAR EQUATIONS 473 Example In the case of the system dx Zxl3 dz Y Y Q h m it quot quot y T we have 1 bl quot392 19 0 0 b2 p1 1 so 5 is satis ed Furthermore by using polar coordinates we see that fxy r2 sin 6 cos 9 lt r Vx y2 r and gxy l2r3 sin2 6 cos 6 lt zrz Vxz y2 I W so fxyr and gxyr gt 0 as xy gt 00 or as r gt O This shows that conditions 6 are also satis ed so 00 is a simple critical point of the system 7 The main facts about the nature of simple critical points are given in the following theorem of Poincar which we state without proof Theorem A Let 00 be a simple critical point of the nonlinear system 4 and consider the related linear system If the critical point 00 of 3 falls under any one of the three major cases described in Section 60 then the critical point 00 of 4 is of the same type As an illustration we examine the nonlinear system 7 of Example 1 whose related linear system is l E 2x3 dt y d 8 2x dt yquot The auxiliary equation of 8 is m2 m 1 0 with roots 1 3 j 2 m1rm2 Detailed treatments can be found in W Hurewicz Lectures on Ordinary Differential Equations pp 8698 MIT Cambridge Mass 1958 L Cesari Asymptotic Behavior and Stability Problems in Ordinary Differential Equations pp 157163 Academic Press New York 1963 or F G Tricomi Differential Equations pp 5372 Blackie Glasgow 1961 434 mwFEmT1aarrmL EnummcrH5 SincB mesa EDDIE am conjulgate mmpiex but nut pure imaginaryF WE have Case C m391dV the Cr i cal paint ED if the linear system 8 is a SpifraL The rem A u critical p0int 39U nf the mnlime a4r system A is M57 3 spiral ill silmuld be uxndeerstmd that whi e the type 0f the criIquotz39icaI paint the same far 4 as it is for 0 in the gases cmrrered the tI1em39em the actual appearance if the paths Amay be s0mEwhat di Ere4mt Fm exyjample P 7396 sh w5 3 typmal saddle ptint fur a linear s3rstemi wheraas 33 suggests haw a nr inear saddle p iI391I l kII ccrtain amnunt of distvmquot tinm is Clcar ly pArc5 ent in the latter but neEverthelEs S the qualitative features f the tws cnn guratinns am the same It natulral tic wander apbnut 1ha EWU borderline ltases which are mm mermined in Th emem W W facts are these the related linear system 3 has a Fd if i made at the nrigjn Casa than the nVunIVinear system iii can have either a rmde or a sp iral and if 3 has a enier at the iI igil l Case E than 4 can have eith er 3 IEELHIEI nr a spiral F01 examplve 0390 is a criIiai point for each of th amp nor inear syrslzems 9 6 6 S r f 39 39 v quot 6 n NONLINEAR EQUATIONS 475 In each case the related linear system is dx dt 39 EX dt It is easy to see that 00 is a center for 10 However it can be shown that while 00 is a center for the rst system of 9 it is a spiral for the second We have already encountered a considerable variety of con gura tions at critical points of linear systems and the above remarks show that no new phenomena appear at simple critical points of nonlinear systems What about critical points that are not simple The possibilities here can best be appreciated by examining a nonlinear system of the form 2 If the linear terms in 2 do not determine the pattern of the paths near the origin then we must consider the second degree terms if these fail to determine the pattern then the third degree terms must be taken into account and so on This suggests that in addition to the linear con gurations a great many others can arise of in nite variety and staggering complexity Several are shown in Fig 84 It is perhaps surprising to realize that such involved patterns as these can occur in connection with systems of rather simple appearance For example the three gures in the upper row show the arrangement of the paths of y 10 X dx dx 3 dx i 2xe2 4 dt Zxy dz X y dt X y W dr 2 2 dr 2 3 60 dt y 2 dt 2xy y dt y4xVxyl In the rst case this can be seen at once by looking at Fig 3 and equation 38 We now discuss the question of stability for a simple critical point The main result here is due to Liapunov if 3 is asymptotically stable at the origin then 4 is also We state this formally as follows Theorem B Let 00 be a simple critical point of the nonlinear system 4 and consider the related linear system 3 If the critical point 00 of 3 is asymptotically stable then the critical point 00 of 4 is also asymptotically stable 39quotSee Hurewicz op cit p 99 pK L11F EMsr4TrAuEQAum IttrN s FIGURE Psrmaf p T39I1e nrlteim 1Am irli SLIEICE S tn cnnstrumit a 5uitab Ie LisapuVn mr Jnc 39ti0 n for the 5ly5tem 4 and this is what we aziim Themcm MI B tc s L15 that the Er g ilti nt of tha linear system 3 sa sfjr the EU39nUWIDES p e aria bl U and q nub v u U M Haw die m Exjyj E Ebxy by prutting H E 4 ab ElihuJ b E KtTII ifg bqbg f W arnzdi 39ib1 1531 p f D wh r 19 x Wm 4 bi39 1 b2 39 2b39Ii NONLINEAR EQUATIONS 477 By 11 we see that D gt 0 and a gt 0 Also an easy calculation shows that D2ac b2 a b af bf a l b a bfab2 a2b ab2 a2b2 aa2 bb22 a bi af bfab2 a2b 2ab2 a2b2 gt 0 so b2 ac lt 0 Thus by Theorem 61B we know that the function E x y is positive de nite Furthermore another calculation whose details we leave to the reader yields SE BE 5alx bly 8 y39a2x 39 bzy 3 J52 b 12 This function is clearly negative de nite so E x y is a Liapunov function for the linear system 3 We next prove that Ex y is also a Liapunov function for the nonlinear system 4 If F and G are de ned by Fxy arx bry fxy and Gxy azx by gxy then since E is known to be positive de nite it suf ces to show that 8E BE F 13 3x 8y 0 is negative de nite If we use 12 then 13 becomes x2 yz ax byfxy bx cygxy and by introducing polar coordinates we can write this as r2 ra cos 9 b sin 6fxy b cos 6 c sin 9gxy Denote the largest of the numbers a lb c by K Our assumption 6 now implies that fltxygtllt5 and lgltxygt ltg for all sufficiently small r gt 0 so 2 2 5a F8 Glt r24Kr lt0 8x ay 6K 3 The reason for the de nitions of a b and c can now be understood we want 12 to be true 473 ii 39FFEREHTIAL Ewumriniqs fuzir th aE H5 Thus is 3 positiir39e de nit i i1miiUn with the pi39ipeit3v ihati 13 i5 negative de iiit3 Themarem 61A imw i139i1piiE5 that i ji is an asympitmtiica11y s39tafbi e rziitical mini Di 4 and the ipmof is EDiI13 iEtEi Tn iiiustmtie this thiwrem we agaiiini unnisiszier iiie l391iii1 E T zayrsteim T if Ea3 Ii239i391Pi6 1 wimisie related liiriear System is 3 Far 3 we have j 1 22 U and 1 E D 50 the critirsai point U is a3ympt iiciaily siabile bath fr the iinezar swiiem 3 and fur the rmiiiinear syst m 39339 Eiiamipile WE kimrw from Scctim1 58 that iii Eqii ii n of minitinn fur the damped vibraiinn5 of 3 pii1ili1 luiin is d1x u39c139xg39 0 5111 I j dc an d a Wi39lEfI39 E is 3 p0is iiiw cUrn5tant The eqiuiva1enit n nliineai quotSt im is 143 15 and since CW i5 eviideniiiy an imiiatcd icriiticali pnini nf the r iatELi iifl39lE i39 sygtem E16 it fiuiiiumws iiiai D50 a siimpie criiticaji piinI mi 15 Inispacciinn sh ws pi Tim T D and q gin 3 U ihat 00 is an asympmticaliy stable criiiicai paint of 16 50 by Thmriem B it i5 3150 an asymptnticaiiy sIaiae miiicai 3iDiI1 Ii if i5 This rtiiecia the Diwirusi iwsircai fat that 16hc pendulum is siigiiiiy diSI39uI39btiCiT than the res uiting rrinzjtiinn will die EiI witih the passage f time NONLINEAR EQUATIONS 479 PROBLEMS 1 Prove that if 00 isquota simple critical point of 4 then it is necessarily isolated Hint Write conditions 6 in the form fxyr 6 gt 0 and gxyr E2 9 O and in the light of5 use polar coordinates to deduce a contradiction from the assumption that the right sides of 4 both vanish at points arbitrarily close to the origin but different from it 2 Sketch the family of curves whose polar equation is r a sin 29 see Fig 84 and express the differential equation of this family in the form dydx Gxy F Ly 3 If 00 is a simple critical point of 4 and q ab2 ab lt 0 then Theorem A implies that 00 is a saddle point of 4 and is therefore unstable Prove that if p a b2 lt O and q ab2 azb gt 0 then 00 is an unstable critical point of 4 Hint Adapt the proof of Theorem B to show that there exists a positive de nite function E x y such that 8E 8E ax by azx bzy x yi ax By and apply Problem 614 Observe that these facts together with Theorem B demonstrate that all the information in Fig 80 about asymptotic stability and instability carries over directly to nonlinear systems with simple critical points from their related linear systems 4 Show that 00 is an asymptotically stable critical point of dx 3 Hi T T X dy 3 4x1y dt but is an unstable critical point of I dx 3 l X 1 y d ampxy3 How are these facts related to the parenthetical remark in Problem 3 5 Verify that 00 is a simple critical point for each of the following systems and determine its nature and stability properties dx d 1txy 2xy jC x y 3x2y a dy rt dy 7 39 3 2 quot quot2X 4 39 dt 2xy y dt yys1nx 6 The van der Pol equation x dx E2 ux2 1Bx0 D DIFFERENTIAL E U39 uTI NE i5 BqUiv3IBnE ID the 535t Em amp I 1 sr39 E W 39l71rwBs39t igite the stab irty pp Ir up rtie5 0f the rst lziicai pcnmi my f I the cases pt 0 D an tE O 1 11 63 N NL14jL AR MECHANICS CNSERVATIVVE SYSTEMiS It is w al krmw39n that energff is di55ipaIedA in the Htinm nsf ally real dynamical sjystam l15L1 N mmugh same fnim Elf frictian H QwE3vcr in certain 5itua39tims this dissipation so sl aw that it can be negl cted oivver IreEativeIy sh rt peyriuds of Iimew In 5uhh cases WE assume the law Df CD39T1S ErV Ii Uf rnErgy4 namely thatV the sum of th kinetic enVErgy and the pmentiall energy E5 mn3tant A systerng of this kind is said to be mnsemarf ulte Thus the urnIating earth can be c0nsiidVerE d 3 c nservative 5yse1Em mum short in tVErvals inf timm in39 mving nl r a few cEntm iE3 but if we want m studfy its Vbcllavinwr I hmugh0ut39 milliTmn5 Of jpECE31T39S WE must talc intr rICD uIJ1 Vtha di55ip1ti1n of ein rgy by tidai frict4irn The simplest c n5er vaIivamp sjgstem C ns i5rts M a mass m attached In a and mn virmg in BI straigfm ling thr3ugh 21 vacuum If 1 demtes the displacement f s fmm H5 EiZ1iLllHif i1fllI lquotl pu5it ic1nh and the reElttu1ri4ng fume Exert ved an m by the spring is r 1 whers k E than we knew that 111 equatimn msf muJtiLJn is 4 kl dr t smiing of lhi p is call Ed n c1M39pr ng39 bampca39u5e the i 39E31 Iffi ng fmnze 15 a fhiwrmar Vfun cAti mn Uf o If m mmve5 through 3 ramting madiuim and the rE3i5tanEe r damLpmg fZMquotC l EI ft d an m is cdrfdVI where C 1 0 than the Equatitm Luvf muti n Of I ID gtEfJ13ET 39 Zifquot a39E system p dgx dx V p FQ39 T quot3 4 I0 U 4 air l iVerE we have mear damApi1r1g becau5e the cJampir1g farce is ac rin ar functiLm anf Mr B ana mgy if I am d f are arbitrary fum39ti nmsa with tihe prupEr4ty that U and 1 x k the mmiE genE1all eqVuatiun d3 d1 P a W NONLINEAR EQUATIONS 481 can be interpreted as the equation of motion of a mass m under the action of a restoring force fx and a damping force gdxdt In general these forces are nonlinear and equation 1 can be regarded as the basic equation of nonlinear mechanics In this section we shall brie y consider the special case of a nonlinear conservative system described by the equation 2 mfi fx0 2 dt in which the damping force is zero and there is consequently no dissipation of energy Equation 2 is equivalent to the autonomous system g y dt 3 IfS 2 dt m 39 If we eliminate dt we obtain the differential equation of the paths of 3 in the phase plane dy fx 3 3 4 my and this can be written in the form my dy fx dx 5 If x x0 and y yo when t to then integrating 5 from to to t yields 1 1 imyz imy 0fXdx Of fhyz fxdx my I0fxdx 6 To interpret this result we observe that myz mdxdt2 is the kinetic energy of the dynamical system and no no dx 7 392 Extensive discussions of 1 with applications to a variety of physical problems can be found in J J Stoker Nonlinear Vibrations lnterscience Wiley New York 1950 and in A A Andronow and C E Chaikin Theory of Oscillations Princeton University Press Princeton NJ 1949 DiFFERENquoti39IAL EGUATi39fNS is its petentiall energy Equaitien 6 therefere expresses the law at eensenratien net energy 1 v q w zx where v 2 iv x i is the censtan t tmtal energy f the system It is elear that 8 is the eqiutaitiein ef the paths ef 3 since we abstained it thy saluting 4 The partieaiar path djtetermined l y speeifying a value at E is a etiwe at enstant energy in the hase piane The eri39tieai paints ef the syrstem 3 are the points t1D where the at are the mats ef the equation fits p6 As we painted eat in Seteitien 53 these are the equi lihriunt paints at the dynarnieai system diesetihed by 2 It is teaident from 4 that the paths eress the I ti at riht angles and are i391iC Jl39iECl i i when they eress the tines xi ti tquot E Etq iii 8 aise shevws that the paths are symtmettrie with respect to the Ztf lti If we write 8 in the farm yJ w vmt at then the paths en he eenstmeteti by the tei aaring easy steps First establish an tzplane with the basis an the same vertical line as the yaxis of the phase plane Fig 85 Nett draw the taph f E PY and sesaerai hetiaemtai lines 5 E in the rs pIane DRE saeh line is sthewn in the gure and observe the geemtetrie meaning ef the di erenee E Vfx Finaiiy for each at multiply E P as ebtained in the pree edi g step by Rim and use fea mttIa P M te plet the eetrtrtespending aah1est ef O in the phase plane aittetetly heietw Hate that since at K the pesitise diteetien aleng any path is tea the tight abuse the t amiss and tn the left heiew this asis EJtaIEttpie G We saw its Seetit39m 58 39t ha39t the equatitJn at nmitien at an tlndampiew peneh1h1t39n is El tti E it stmz U 10 as where tit C a psitive eenstant Sinee this equatiiten is ef the fetrn it ean he iatterlpreted as Lieserihi39ng the tin39Jarnpetil reet iiineat met ien ii a unit mass lJJt lII39i i39 the infitlenee Di a ttettii rIear spring whese 1quoteste139ing fet ee 6 i sint The u ittquottCMquott1Ct11S system eqnis39vaIlent the Mt is Q va t39y J it Si1itJ at til Wm NONLINEAR EQUATIONS 483 Z I I I I I I I I I I I I I I I I I I I I e I I quot I I I I I I I I III I I I I I I 39 I 39 I 39 I 39 I I I E Vm I I 3 E Vm FIGURE 85 and its critical points are 00 lJt0 I2Jl390 The differential equation of the paths is Q ksinx dx y and by separating variables and integrating we see that the equation of the family of paths is 1 Ey2 k kcosx E DFFlLREHTlAL EDUATEU39HE This 0 Ev id 11 jF asf the farm 3s wslmsrks m 1 and u air k k KrI1Ss 1 is thus patEmisl snergyr we nsw wnsstsrust the pssms by r rst drsswiing the graph sf 2 Wxs sud svsvsrsl nes 2 39 ENE th EEipl rE Fig Em where 3 E 2k is the snsnly ns shswn Fmm this we read nff the wrsisusss O VxJ suds sfkcstIh thss spams in the phase p srmea dirsstIy Irmmw by quotusing I39 T 1 39 It is sissr fmn1 phases psrtmit that if the tum energy 0 QT b wiesna DI snisd Es tiheng the susgrrsezsp nadisng paths are cil sedl srnsd equatim W has paerimdizz s rsLlustsi n s On the nt hsur hamML if thsn the path is mt Ems am the mrresp0ns d39ing sol Ln Itin of ID is tram pesi disc The saiiuss P 2k 5EfpEi7EIE5 the was mnssi sf msasstiun and foxrs this rsssnn a psth snrrssp ndisng In x 2k is uitslisad a sspsrs rix s The wavy paths m1 Iisi e the s sparstnTces csirrcspnsn d to w39i1iJrsiisnsg mmi ns nf the pendju Isusm nd the 1lnsrrd paths inside E11 n svI 1i Nstry mDTTimquotl It is 1139idEnE Elma thke tritia paints sure sltsrnstzsljr unstable saddle pninrts and stsbls but rnmt sss39jpsmptmi cam stable uenmrss For the sakas sf r mtrss ti it is imsuasstsirmg In mnsi der the If T 0 SUSS A V Al I p 7 Lu 3 air 1 39 v FIGURE F NONLINEAR EQUATIONS 485 V ll FIGURE 87 effect of transforming this conservative dynamical system into a noncon servative system by introducing a linear damping force The equation of motion then takes the form c srnx dz dx k 39 0 gt 0 dtz dz C and the con guration of the paths is suggested in Fig 87 We nd that the centers in Fig 86 become asymptotically stable spirals and also that every path except the separatrices entering the saddle points as t gt o0 ultimately winds into one of these spirals PROBLEMS 1 IffO 0 and xfx gt O for x 0 show that the paths of y 23 dt2 0 are closed curves surrounding the origin in the phase plane that is show that the critical point x O y dxdt 0 is a stable but not asymptotically stable center Describe this critical point with respect to its nature and stability if f0 0 and xfx lt0 for x at 0 2 Most actual springs are not linear A nonlinear spring is called hard or soft according as the magnitude of the restoring force increases more rapidly or less rapidly than a linear function of the displacement The equation 2 d T kxax30 kgt0 describes the motion of a hard spring if a gt O and a soft spring if cv lt 0 Sketch the paths in each case DFFEREhfTAtL Ee39uaTte39rss Find the E ftlb tthltt set the paths ef ax V A t 39 J 4 113 L as antt slteteh these paths in the phase 39ptEtttet Leeate the etitieat peints and deterniine the ttatute ref eaeh 4 Since quotby EqLtE1 iEtt I 5 we hese dWj dM x the eritieel peittts ef I are the peiitts en the stasis in the phase plane at w hiieh hquotquot39 s ID In tertns eat the curve 2 Vxij if this eurse is sttteeth and wet hehssedmther39e are three iptessihihties maximta I tiI7liI39Iquotli hand peints ef in ettieiL Stteteh all three pessihiti39ti es and determine the type ef eritieat peint asseerietee with eaeh at eritieet peint ef the third type is ealled a cusp 54 Pametmc seLU39t1 eiis THE tPOlNC5ARE BENtIXSDN Censider a nentsineatr aLt t I1Dm lJl5 system 1 dx 1 in whieh the ftinetiiens Fsy and Gsy are eentinueits and hase eentiniieits rst pairtial detisattiseist T hit39tZ39i ghuTZ the phase piatte Utir werh see far has teld us ptaetieaily nething lh ttl the paths ef t eaeept in the rtetighherheed ef certain types ef critical eirnts HUWE iFEtquot in rttatiy pzreiettis we are IT1tllCh mere interested in the glehat prepettiets heft paths than we are in these lees preperties Geisha prepertites et paths are these that describe their hehmtiet ever laiige tegiens ef the phase plane and in genera they are very cti ieiilt it etstehtisth The eentral ptehlemi ef the gtehat theery is that ef detetn1itting whetheir L has elesed pathst As we tetnarked in Seetimi 533 this ptehiem is impettaint heeaiise ef its release eernneetin with the issue ef whether 1 has petieie seltitieiis A selutiten rIft and yft ef 139 is said the he petrieeie if neither fiunettien is censlzarnt heth are defined fer atl t and there exists a nutnher E such that I t l it and quott T pyt fer all t The sntellest T witth this preperty is eatletl the parties ef the stettuitient It is evident thati eaeh periedie selutietn pt 1 1 de nes a elesee path that is ttaversxed ernete as t I1I391CI39Eti39E1E EtE free 3 ate In t T fer any tam 393 Every periedie seh1 t39iien has a periee in this sense Whyquot NONLINEAR EQUATIONS 487 Conversely it is easy to see that if C xtyt is a closed path of 1 then xt yt is a periodic solution Accordingly the search for periodic solutions of 1 reduces to a search for closed paths We know from Section 60 that a linear system has closed paths if and only if the roots of the auxiliary equation are pure imaginary and in this case every path is closed Thus for a linear system either every path is closed or else no path is closed On the other hand a nonlinear system can perfectly well have a closed path that is isolated in the sense that no other closed paths are near to it The following is a wellknown example of such a system dx 1 2 2 dt yx x y 2 dy 1 2 2 To solve this system we introduce polar coordinates r and 6 where x r cos 6 and y r sin 6 If we differentiate the relations x2 yz r2 and 9 tan yx we obtain the useful formulas xt ydy d and Q 95 Zde Z1quot39d xd dzquotE 3 On multiplying the rst equation of 2 by x and the second by y and adding we nd that rSE r21 r2 4 Similarly if we multiply the second by x and the rst by y and subtract we get r r 5 The system 2 has a single critical point at r 0 Since we are concerned only with nding the paths we may assume that r gt 0 In this case 4 and 5 show that 2 becomes 5 dtr1 r2 6 Cig1 dt 39 These equations are easy to solve separately and the general solution of mFFEREN391 iAL EDiUATEDH5 the sy5teLm E fnund to B The c0rreL5gpDn ing general 5UluVlicm of 2 is ms 5 rn x r N Let 115 iaAnai1s xze 7 genimetr4icalIly 88 If 2 0 we have tbs smlutsiuns r 1 an E If 9 Awihkh tmrise nut Ih BIDSEd tir cuar path Ax y393 1 in the r0untErcrInVckwis e d irectimn If a 2 0 it is clear that r I 1 ma timt r 1 as I 0 W Aim if T 2 C WE see that r L and agairI r 6 1 as 1A Th sa DbsenratiVnns show that there exi51I5 a Siillgl rELlD5Ed path r ll wThich all what pathsa appfmah spzirally from tha u t5i4dje car the inside as r m Fell NONLINEAR EQUATIONS 489 In the above discussion we have shown that the system 2 has a closed path by actually nding such a path In general of course we cannot hope to be able to do this What we need are tests that make it possible for us to conclude that certain regions of the phase plane do or do not contain closed paths Our rst test is given in the following theorem of Poincar A proof is sketched in Problem 1 Theorem A A closed path of the system 1 necessarily surrounds at least one critical point of this system This result gives a negative criterion of rather limited value a system without critical points in a given region cannot have closed paths in that region Our next theorem provides another negative criterion and is due to Bendixson 14 Theorem B If 3FBx 8G8y is always positive or always negative in a certain region of the phase plane then the system 1 cannot have closed paths in that region Proof Assume that the region contains a closed path C xtyt with interior R Then Green39s theorem and our hypothesis yield 1 8F ac Fd Gdx U dxd see 1 y R ax ay y However along C we have dx Fdt and dy Gdt so T LFdy GdxL FG GFdtO This contradiction shows that our initial assumption is false so the region under consideration cannot contain any closed path These theorems are sometimes useful but what we really want are positive criteria giving suf cient conditions for the existence of closed paths of 1 One of the few general theorems of this kind is the classical Poincare Bendixson theorem which we now state without proof Ivar Otto Bendixson 18611935 was a Swedish mathematician who published one important memoir in 1901 supplementing some of Poincar s earlier work He served as professor and later as president at the University of Stockholm and was an energetic longtime member of the Stockholm City Council 5 For details see Hurewicz loc cit pp 102111 or Cesari loc cit pp 163167 D1iFFEREH MAL Eafr39Lsquotrm ar T1EEil E39m Y LET R be H miuinrieAd regmn 3 Athg phase pmnEA rQg39E1rhiE39r with as beundary and m gumg mm R dae5 ram rfm a39in any criiicai pm n Elfquot the syggiem quot1 0 I E I l H path of that Lies in R fm some and r mmim in R fr a39L3939 I 33 in then C Eir her E aF ImiEd paw M y Epir ii mward E tinged Vpapth Q5 I as in EiI5IE F case the 3y3r m El has a timed pa rh in pm In mdgterA ms mndnr5tand this tatememt wait was cunsidcr Ithue 5ituati n 5Ug SEd p I9 Hera R c imsist if the mm dashgd rurves tagefthe4r with the rinAg shVaped regi n between them SLIPPGSE that th ventJr pnint5 inm at ever bnunadgary PCri j1Z Then eveIy path thmujgh a bmmdary pu irm at 2 mg rnwst enter and can n everV leave it an1i under mhescs circum5tianc amps the tlmw eVm assms that musrt spiral Inward a ELIUf d e We Ahave cihnaen 3 rimg5haaped regi n R Ellrustrata the 1he Iem Vbertause a cWn5 Ed path lJill e Pdz mu t 5um1und 3 criticaTl pint P in the ugureJ and R p exch1dE all Vc139ilical1 pDimts The 5y5tem EV pmvzides a si4mpleA ap plicatAim1 at theme i deas M clear that 2 v a critica1 mint at I D and alsn that the mgi n R IWEJEWJEEII the ireles r j and r 2 am1atains rm crit mal paints In u eaurl iampr arT1amgr5is we f tuund that r1 3 rm r 3 0f FIG B9 NONLINEAR EQUATIONS 491 This shows that drdt gt O on the inner circle and drdt lt O on the outer circle so the vector V points into R at all boundary points Thus any path through a boundary point will enter R and remain in R as t gt 00 and by the Poincar Bendixson theorem we know that R contains a closed path Co We have already seen that the circle r 1 is the closed path whose existence is guaranteed in this way The PoincareBendixson theorem is quite satisfying from a theoret ical point of view but in general it is rather difficult to apply A more practical criterion has been developed that assures the existence of closed paths for equations of the form 2 22 ring gm 0 9 which is called Li nard s equation When we speak of a closed path for such an equation we of course mean a closed path of the equivalent system dx tgt 33 gm my and as we know a closed path of 10 corresponds to a periodic solution of 9 The fundamental statement about the closed paths of 9 is the following theorem 10 Theorem D Li nard s Theorem Let the functions f x and gx satisfy the following conditions i both are continuous and have continuous derivatives for all x ii gx is an odd function such that gx gt 0 for x gt 0 and fx is an even function and iii the odd function F x I3 f x dx has exactly one positive zero at x a is negative for 0 lt x lt a is positive and nondecreasing for x gt a and F x gt 00 as x gt 00 Then equation 9 has a unique closed path surrounding the origin in the phase plane and this path is approached spirally by every other path as t 00 For the bene t of the skeptical and tenacious reader who is rightly reluctant to accept unsupported assertions a proof of this theorem is G Alfred Li nard 18691958 was a French scientist who spent most of his career teaching applied physics at the School of Mines in Paris of which he became director in 1929 His physical research was mainly in the areas of electricity and magnetism elasticity and hydrodynamics From time to time he worked on mathematical problems arising from his other scienti c investigations and in 1933 was elected president of the French Mathematical Society He was an unassuming bachelor whose life was devoted entirely to his work and his students 492 etereehetnmtt ete LmTteHe gitten in tAppetttd1ixt An 39irttttiti ve utndetr etemrding ef the fete of the hytptethteees can the geihe d tthinking hf 9 in terms ehf the ideas ef the prewietttss sheetiera Frm this peiint eif vtiew equettietn 9 is the equettietnt of mmien ef at tmiit mess tetteethedt to e and subject tie the duel in uence ef eh 39FES39t D139quoti g fereret gx and 3 damping fertee ex dxfdtt The taeeumptittetn b titltfl 33 amertthte ten sejyintg that the S39pt39quotilg eets est we weemid EIIEECEJ and t d twe tdtiminieh the magntittude of any tdispteteementt Oh the etthter hand the aseumptierte ehheut f x7 r39eughtlyt that It is htege39tive feta emrah M and quotLp 539i1tilquot39E fer llerjge Mammteen that the Iquotquot t 39lZiliIIiE 1 is intenei ee flevr ernetl rf and t39Et IEd fer large truly an ihEtquotEfDE tsends the eetttle dewn ihte a eteadiy eteettiltetietn This rether peeutierr bethtte etr of f een alzse be mtpreseed by seyringw that the pthyeieett system aheerbe energy wheh lei ie eme and dise jpetes it when t is llierge The mein epptIhiee39ti e h ef Litenarwrs theezrem P X to the tt ti der Pet eque en ll dgx L dx e V V fair IU U J U 11 where pt ie eeeutmetd the be e peeititte teetnetent fer physieeI ree5met Here ftfxt 2 t1x3 1 end I E 1 EU eencli39titett i is eletarly settte edtt It is eq ttaW hear that C Jt1Id39i ft0 P N ts true Since N G K W L I VIII J 3 we eee thet jejt has 3 singte pUSit l7IVE zete at 1 E is ftBg3sti irB fer D 39 5 11 V3 is peeittitre tfer J 2 P and 39that Ftf Ha e es 1 2 D Fitntetljtyi e 2 ftIE 1 G pestittve fer x 151 ee Flf x E etetrtatinly ntentdtesereaeting iin feet itnt ert eteeing ftZ1 139 1 e E Aectertdtitngly all the tErE3ndili1 t39tS et the 1ChErUI EII39t t are met and we eehelude that e quettet1 G11 has a utttitqtte elteeed path per1ia etdire eettttiiert that is tapprettaehted epiralily f tsyII1p39IDI iE39E1 by every ether path I1EIZE1Iih i3w seluttitetnt PROBLEMS 1 A jpreef elf Tthtetetem A eeh he built en the tfieltetwing ge nte39trie ideas Fig 90 Let C be e sijm ple eleeed curve tmt neeteeeeritI he path in the phase piJene and eeettme that C ees net piase threugh asrty ereitieej peintt ef the ejeteltrm ft If Beltheeat Wan deerquot Punt U3iE9 A1959 e Duteh 5G7iEiquott39li5t epectaAit iteing in the tZheetrettieel eepeE Ie etf ttttdinwequotmgitrteet ittg inmetted the ett ttd3 hf E39qu3 t iEiI39t 11 in the 19295 and thereby 5tiITt l Ll1 IiE39 t2I3 Ltenazrd end ethere te ht 39e39eetigete the matheme tieel therery ef ee l39I1eus39teined eeetiiLIte titne in ntet39t1iI 1eEur meehenieet uJa NONLINEAR EQUATIONS 493 V41 5ll FIGURE 90 P xy is a point on C then Vxy Fxyi Gxyi is a nonzero vector and therefore has a de nite direction given by the angle 9 If P moves once around C in the counterclockwise direction the angle 6 changes by an amount A6 Zmz where n is a positive integer zero or a negative integer This integer n is called the index of C If C shrinks continuously to a smaller simple closed curve C0 without passing over any critical point then its index varies continuously and since the index is an integer it cannot change a If C is a path of 1 show that its index is 1 b If C is a path of 1 that contains no critical points show that a small Co has index 0 and from this infer Theorem A Consider the nonlinear autonomous system dx Ci4x4y xx2y2 d 2 4x 4y yx2 yz a Transform the system into polar coordinate form b Apply the Poincar Bendixson theorem to show that there is a closed A path between the circles r 1 and r 3 c Find the general nonconstant solution xt and y yt of the original system and use this to nd a periodic solution corresponding to the closed path whose existence was established in b d Sketch the closed path and at least two other paths in the phase plane l 494 DiIFFEaamraL EauamiNa Shaaw that the rmann39Iinaatr a39uataraaa39maua ayastam E 33 3 a39aquot L quotE 7 9 I 7 lEz3jquot has a pmriodiic a a39Eatian Ia aaah nf the fafialawaing rsaaaa use a th auram of this sarztiaen ma damrmiaa whathar Ur nm the given diffEi EJ ti 1 Eq11 ti has a aparazadaiaz 5al1mtian ut r a it E E 175 dlx ml rar K 2 V 1 E quot39 d2x it dxj a fab 39 2 1 15 Fl M 394 v 3x391E a F ax an E P I 5 1Ed a dr a39r aft Shaw am diffaraatiai awaataiaa af tha fiarm EH d a mi Ma 1 Ex 13 am m E H can be 39Eraa5fa rmad imam the van dar PtI aqaatiana Pay a change M the indapandaat39 vaariabla AafPPEN DK a P0iNC ARE aoaniaiedl at Ella Jawlaa Hsanrfi Pniacar 154 1I91E univerasaIaly 39bagiJnninag uf the Izrweantiath an39laur s as the graarmesl mathama1Iiciaan at his gaaaraatimua began his academic career at Caaa in 1879a but oanlgr rcwa wars later he was aappaaintad I20 a p rafcamraaahip at the Sarhmma Ha ramaaimada thagra fur 1116 I657 DE his lifa Eactuar39ing an a diaaram auhjacta ea111 year In his Ieatauaara5awl1icah waraa aditad aand published lay his a1udantaltl1a taraatad with great ariginaiitjy and nmaatery arf tacfihniqua wirtualalya all k arwna elds af pure and appliad n1athamatiaaaa and maaya that awara am lmawn until he diaaaaavaraa 1l1am Mtaagathaar ha jaraducad mare than 30 tazaahniaal banks an n1athEamaticaE physaicia ajad celestial macaahaanica halt a d aaan lmaiks at a mare ppulaar gnatmra and alamast SUU BSxE EIzIquot39Ch paars cm mathamraIiEs Ha was a qauiac1v 3 paawarfula and raa aeaaaa tahiakara nut given ta Iiaagamn uwam39 ataila anti was described I313 DIIE aif his caantarmapnraaiaaa as a rannqaerarJ am a alnai5I 11 3150 had the advaaantaga of a prndigiaus maraagryu and haihiataally 8 his maathaaamamzzs m NONLINEAR EQUATIONS 495 his head as he paced back and forth in his study writing it down only after it was complete in his mind He was elected to the Academy of Sciences at the very early age of thirtytwo The academician who proposed him for membership said that his work is above ordinary praise and reminds us inevitably of what Jacobi wrote of Abel that he had settled questions which before him were unimagined Poincar s rst great achievement in mathematics was in analysis He generalized the idea of the periodicity of a function by creating his theory of automorphic functions The elementary trigonometric and exponential functions are singly periodic and the elliptic functions are doubly periodic Poincar s automorphic functions constitute a vast generalization of these for they are invariant under a countably in nite group oflinear fractional transformations and include the rich theory of elliptic functions as a detail He applied them to solve linear differential equations with algebraic coefficients and also showed how they can be used to uniformize algebraic curves that is to express the coordinates of any point on such a curve by means of singlevalued functions xt and yt of a single parameter I In the 18805 and 1890s automorphic functions developed into an extensive branch of mathematics involving in addition to analysis group theory number theory algebraic geometry and non Euclidean geometry Another focal point of his thought can be found in his researches into celestial mechanics Les M thodes Nouvelle de la M canique C leste three volumes 1892 1899 In the course of this work he developed his theory of asymptotic expansions which kindled interest in divergent series studied the stability of orbits and initiated the qualitative theory of nonlinear differential equations His celebrated investigations into the evolution of celestial bodies led him to study the equilibrium shapes of a rotating mass of uid held together by gravitational attraction and he discovered the pearshaped gures that played an important role in the later work of Sir G H Darwin Charles son 8 In Poincare s summary of these discoveries he writes Let us imagine a rotating uid body contracting by cooling but slowly enough to remain homogeneous and for the rotation to be the same in all its parts At rst very approximately a sphere the gure of this mass will become an ellipsoid of revolution which will atten more and more then at a certain moment it will be transformed into anellipsoid with three unequal axes Later the gure will cease to be an ellipsoid and will become pearshaped until at last the mass hollowing out more and more at its waist will separate into two distinct and unequal bodies These ideas have gained additional interest 8 See G H Darwin The Tides chap XVIII Houghton Mif in Boston 1899 DEFFEREHTIAL eewrnees in em ewe time fer with thee aid ef erti eiel satellites geephyeieiete have EEEnIl found that eeretih itself sleightly peeereheped Menjy ef the prebieme he en eeuneteeredT in this peried were the seeds ef new ways eff theeinking wheieh have grewne and eurieheed etweeetieeetehee eenetury metheemeties We have already mentieer1ed divergent s er39ies and nenigineear di eremiel equetiens In edidietiee attempts to master the qiusaleitetivee nature ef euwes and surfaces in dimeensieanal spaces r eeulte in hie fameus memeir Anelyeeis eirm 18 ewhieh meet B339EfpEl39t5 aree mmfke the legeinnieng ef the medern era in elgebmee tepelregy Also in his seteudy eef peeriedieeee erbitst he femdede the subject ef teepeljlegieel er queitetivltee djmemies The type ef methsernetieal pvrlz em mart arises here 0 iiluelreaied by e theeeereme he e emjeeeeturlteedi in 1912 but diid net me te prequotwe if e eeeeme eene eeeneteieneeeuze tree 1sferemeteieen earriee the ring hevunrdeda by twee eeene eeezritrice circles inete i ES Ilf in such a way as to presewe greens and to mmne the pyeeints ef the inner eirele eleekwiee and these ef the outer circle eeunteereieekwiee then at least twee peinets must remerin xed This etheerern has imepevrtent eppliieetieer11s tee 3r eleeseieel epreblreme ef three belies and aflee re P3r meetiee ef ea hiilierd bell en a eeenweex ebi39llierd table A preecf was fume in e1 I lfe3 by Birkhee a yweueng Ameerieen mathemeteieeien eAneethere remerkeeehEee dis ee very in this eld new knewn as the lP Urm cear39e reeurretlee tiheereeme relates be me lengrange behevier ef een erva tiwe dynemieall 5yetemeg p d 1reeLIlt seemed to deeemenstrete the futility ef eemeempeerergx eeffeeerte te cledrueee the seeend law ef thermee dynamieee frwem eelasmeale emeeheneies and the eneeuieng eeeen trequotverejy was the hietereieeegl eemree ef medem ergeme itheery One of the meet striking eff Peeeien ceere s mam eeetributiene m methemetieeiel physies was his famous paper ef 1906 en the d5memieee ef the eleeeetrenm He had been thinking ebeLII the feund etiene ef physics for men rltee139e end iendepende ently Elf Einsterin hedf ebteineede men3r ef the FE5iL1ltE emf the epeeeiel theery ef relativity39E39quot The main vdii ereenee was that Eeienesteeirte treatment was based en elemeenteel eiees ereelaitieg tee p eiignefls while Pereineear e s was funedede en the theory ef eIeLe1hreeemegnetiem and wee theerefVer e limite d in its K eP39iP1iEE1h ity me phenemeena eseeeiauteedquot with this theeiryz Feeinceare had e high 4 fer Einseteien e abieleitieei end 0 2 1191 1 reeemmen deed 5 fer p 7 ret aeadiemic peeiteie H See 1 Birkh e JyzeltmEeef E wremee ehep V I amerieen Metthsemaeetieem Seeietj Cellemlium Pehi39ieet iI1 n5e eel IX Prwvi eenCe 1 I921 3 ei5eeeeien ef the hieterieeIe hzaekgreund is given by Cher39Iee Seiri39hner J39rH Henri Peineeure end the P raeiple ef ReLleIeirEt393r39e Ami J Phys Vel P p er 3 See z Uneeln Seheueter ed 39Treesury ef the 11 er EeE39e Greed LeErere E Sigmen end SehueIeer New fem 19e u NONLINEAR EQUATIONS 497 In 1902 he turned as a side interest to writing and lecturing for a wider public in an effort to share with nonspecialists his enthusiasm for the meaning and human importance of mathematics and science These lighter works have been collected in four books La Science et l Hyp0th se 1903 La Valeur de la Science 1904 Science et M thode 1908 and Derni res Pens es 191322 They are clear witty profound and altogether delightful and show him to be a master of French prose at its best In the most famous of these essays the one on mathematical discovery he looked into himself and analyzed his own mental processes and in so doing provided the rest of us with some rare glimpses into the mind of a genius at work As Jourdain wrote in his obituary One of the many reasons for which he will live is that he made it possible for us to understand him as well as to admire him At the present time mathematical knowledge is said to be doubling every 10 years or so though some remain skeptical about the permanent value of this accumulation It is generally believed to be impossible now for any human being to understand thoroughly more than one or two of the four main subdivisions of mathematics analysis algebra geometry and number theory to say nothing of mathematical physics as well Poincar had creative command of the whole of mathematics as it existed in his day and he was probably the last man who will ever be in this position APPENDIX B PROOF OF LIENARD S THEOREM v Consider Li nard s equation 72 fltxgt f gx 0 lt1 and assume that f x and gx satisfy the following conditions i fx and gx are continuous and have continuous derivatives ii gx is an odd function such that gx gt 0 for x gt 0 and f x is an even function and iii the odd function F x fi f x dx has exactly one positive zero at x a is negative for 0 lt x lt a is positive and nondecreasing for x gt a and Fx gt 00 as x gt 00 We shall prove that equation 1 has a unique closed path surrounding the origin in the phase plane and that this path is approached spirally by every other path as t gt 00 22 All have been published in English translation by Dover Publications New York 493 l1tIFFEREHTlhL Et JLJ tTtEthi395 The ejyttettm equtitveltentt te in the yrheee plane is 2 39tU fUMa By eendtistteht i the basic theetem en the texieteeee end uniquen eest ef teehttitens thetds It tetlerwe fmm eettdttiettt ti tthett g D and g t39 B 0 far 1 1 en the erigttt is the etnly erittcet ptitnlzt Meet we knew that emf etese rhust Sailtlititttttd the UVEaiE ea tI The feet that d2It39 p t t quot E t L f it eFen erLtggteets itttt etduetitnth e new quotvariebtet 5 n Fx l thte netetitt E ttr iIiDnt 1 is eqttivehtent ten the ts3retem 3 sin the H tptentex Again we see that the ext5tenee attd eniqfuene55 l3h IEm htettltds that the D39Iquottgi the enly etitieat pettmw and that any closed peth mm 5urr39emtd the erigtha Tthe lenetetmine teerreepetndtenee IJv39 pA eye between the paints ef the tgwe planes C eentirtueuts heth weyeg se Cl 5EI i paths eetrampesipend te etesetd peth5 and the tunt guretiDtn ef the tn the two vptenes are quteltta ttive y tsimitar The dti terettttiet equtattien ef the Apetths ef 3 35 err JE tz tFt These peth5 ete teeeitett the analyze then 39ITE t39Eil39 eetttresapendi1tt paths in the ph sE plane fete the tfet l wtmg teesetne Fif f tinee heath gt and Fm p ed teqttetiens 3 and t ere 39llt ClquottEttgEd when 1 and z ere replaced Vb 3 and 5quot31 This means that enfy CquotLt1quott39E tjtmtt1et tie ta 3 petth with testzreet the the 39UI39igji p else a path T htus we knew the paths in the rtightt hetttfplane x U theee 0 the left h fIPIa mE gr 1 U eren 2 ehtaitted at enee iby te eetien tviemugth the tttetn NONLINEAR EQUATIONS 499 Second equation 4 shows that the paths become horizontal only as they cross the zaxis and become vertical only as they cross the curve 2 F x Also an inspection of the signs of the right sides of equations 3 shows that all paths are directed to the right above the curve 2 F x and to the left below this curve and move downward or upward according as x gt 0 or x lt 0 These remarks mean that the curve 2 F x the zaxis and the vertical line through any point Q on the right half of the curve 2 F x can be crossed only in the directions indicated by the arrows in Fig 91 Suppose that the solution of 3 de ning the path C through Q is so chosen that the point Q corresponds to the value I 0 of the parameter Then as t increases into positive values a point on C with coordinatesxt and yt moves down and to the left until it crosses the zaxis at a point R and as tdecreases into negative values the point on C rises to the left until it crosses the zaxis at a point P It will be convenient to let b be the abscissa of Q and to denote the path C by C It is easy to see from the symmetry property that when the path Cb is continued beyond P and R into the left half of the plane the result will be a closed path if and only if the distances OP and OR are equal To show that there is a unique closed path it therefore suf ces to show that there is a unique value of b with the property that OP OR FIGURE 91 50039 DIFFIE1ENTmiLE UATl Ns TU gprtnzwra thh iia we intrmu ce Gar E I gax rm and considcr the functimm Exz 211 Gx whri ch reduwzzes ta 2372 cm the Ei V quotJ 39SIi VA inAng any wars have dE39 gt air P H I M 51 Aaquot L H I his I 50 pg FEEL If we mmpute the line int egraI of aiung the path from P to we mbtai n b I Fig 7 J ER EF ORE OPE PH PH 3 50 ii suf ces to show that there is a unique sfuch mat Kb a b at tihen p and dz are 4n egati1E 50 1quot U aI1d E caImrrt be 10sEd Slilpp s 110w that b 3 an as in Figg E We 5lit Ib into two parts rgm mg Fa mb 2 Fdz P5 TR 5 so thai 0 Since F and dz iare uwgati39ve as z is traver3e d fmm P I10 and Tram T tn R3 it is cfibear that quotlb1 E U Om the azzwther ham was gm fmm 04 tn Y aI nng Cb we have F 0 and dz I L 50 J73b r If Dur iimmtdiate puir puse is to EI39lL39w that Hgbvj is a derea5i4ng function 01 b by saparately mnid ering quot1b and fg b we nate that equation enable5 us tr wrime b a39z j HgrF it it Fr The zef ersi uf in reasVi11ig is to raialse the arc and tag Imwer the are whiizh detraaAsles the magnitude f VgxAF z F x M a given I hetween D and Hi 3 iI1E1Ea the limits Bf intBAgrati0n faJr Mb are xed the NONLINEAR EQUATIONS 501 result is a decrease in I1b Furthermore since F x is positive and nondecreasing to the right of a we see that an increase in b gives rise to an increase in the positive number I2b and hence to a decrease in I2b Thus Ib I1b 12b is a decreasing function for b 2 a We now show that I2b gt 00 as b gt 00 If L in Fig 91 is xed and K is to the right of L then 121 Fdz lt I fdz lt LMLN ST NK and since LN gt 00 as b gt 00 we have I2b gt 00 Accordingly Ib is a decreasing continuous function of b for b 2 a 1a gt 0 and Ib gt 00 as b gt 00 It follows that Ib 0 for one andonly one b bo so there is one and only one closed path C0 Finally we observe that OR gt OP for b lt b0 and from this and the symmetry we conclude that paths inside C0 spiral out to Cbo Similarly the fact that OR lt OP for b gt bo implies that paths outside Cbo spiral in to Cbo CHAPTER THE CALCULUS OF VARIATIO 65 pk TYPIACAL The calcu us of variati ms has been Gm Bf the Amajmr branches nf aapnalysiss for more than twn rEE tuf iES It is 3 t rl Uf great power that can be apEi6 ed to a wide variety inf prHEmS in pure matfhAsirnatics It can alga he use my expr ea5 the basic pIinvip1amp5 of matheamatica phgrsim in frrnV5 of the utmost aimlicity and E Bg I l1CE avor 0f the subjeat is ueasr EU grasp by c rn5iVdaring a few f P typtixcal pr b ems Svupp se that WU pDi39l ilIE P and are giw3n a plant Fig Tlmrbe are i n nitVe VyV many Acurws jmini1391 g tiheis pQin1t54V and we can ask which of these cuwas is the shnrtwest Th intuitive answer is uf nJrsc 3 gtraixght ESmmgry can als ask which curve will EneratE Igh surfmzfe Uf rVev0IutiDn of 5m EeSt ama when V39 hFEd abmut the Iaxia and in this case the answer is far fmm aclieazr If we thinkV at a typicVaVl cuwe as 31 1frirtiDVnles wire in a vert4ical FEEMai than Vanmther nontrivial p1r 07blELm is that f ndim the curve down which a head will slide fram P to in the sllmrtgesat time This is the famfus brachistochr nE pI tZ1Ib1EII1 of Jczuhn Bern3uiii which we discusscd in SEICWDH O lntuitivAe amswera 13 such ques tfim391s are quite rare and the ca1 uluS PJL variatim1s mm39idas a u1r1if0rm ana lyrtxigrcai meth d fur daaling with 5ituatiun5 f this kind THE CALCULUS or VARIATIONS 503 Yli all FIGURE 92 Every student of elementary calculus is familiar with the problem of nding points at which a function of a single variable has maximum or minimum values The above problems show that in the calculus of variations we consider some quantity arc length surface area time of descent that depends on an entire curve and we seek the curve that minimizes the quantity in question The calculus of variations also deals with minimum problems depending on surfaces For example if a circular wire is bent in any manner and dipped into a soap solution then the soap lm spanning the wire will assume the shape of the surface of smallest area bounded by the wire The mathematical problem is to nd the surface from this minimum property and the known shape of the wire In addition the calculus of variations has played an important role as a unifying in uence in mechanics and as a guide in the mathematical interpretation of many physical phenomena For instance it has been found that if the con guration of a system of moving particles is governed by their mutual gravitational attractions then their actual paths will be minimizing curves for the integral with respect to time of the difference between the kinetic and potential energies of the system This far reaching statement 39of classical mechanics is known as Hamilt0n s principle after its discoverer Also in modern physics Einstein made extensive use of the calculus of variations in his work on general relativity and Schrodinger used it to discover his famous wave equation which is one of the cornerstones of quantum mechanics A few of the problems of the calculus of variations are very old and were considered and partly solved by the ancient Greeks The invention of ordinary calculus by Newton and Leibniz stimulated the study of a number of variational problems and some of these were solved by etIFFEeerrrtaL Eettlattevas ingtenielua special hmetlhedls HUIWEVBT the euhjeet was launehed as a eeherent biI39 ElCh ef analyai a lhy Euler in 144 witht his dieeeverj ef the basic i ierelnttllal equation fer a 39minitmtial ng carve We shall disease iEt1lter s equlatietn in the met tt section but met we ebsewe that each of the pmbltema deaetibed in the aeeemad paragrelph rf this aeetien is a special eaee ef the fellewitng mere Keneral preblem Let P and Q have ee etdilln alltee x1y1t and ltgyg and C0iFl5ldEl39 the family ef funetielne y air 1 that ealtiafy the heundlary teendltlene y11t M and yfx yg 1hat is the graph ef tl lTlllS jeirl P and 9 Theme we wish te nd the tfunetien in this family that minimiaaesl an integralt ef the ferm My J x2fxTy y an 2 Te aee that this prehlem lndeed entaihgag the ethera we nete that the length ef the eurael l is l M and that the area ef the atlrfaee ef reve lutlen ehtalned by r te39aetlvilng it abut the aaazia la Ph mi dx 3 11 J 2tyl V 1 y 3 dx 4 In the ease ef the eerae ef quielteat deeeetlt it let eetrwenient te itnvett the eeer l39inate aystem and take the paint P at the eriglntl as in Since the speted U dam is given by U V gyx tetatl time ef deseerlt la 39jr 1r FIGURE 93 THE CALCULUS or VARIATIONS 505 the integral of dsv and the integral to be minimized is Accordingly the function fxyy occurring in 2 has the respective forms V1 y 2 2ny1y 2 and V1 y 225 in our three problems It is necessary to be somewhat more precise in formulating the basic problem of minimizing the integral 2 First we will always assume that the function f x y y has continuous partial derivatives of the second order with respect to x y and y The next question is What types of functions 1 are to be allowed The integral 2 is a wellde ned real number whenever the integrand is continuous as a function of x and for this it suf ces to assume that y x is continuous However in order to guarantee the validity of the operations we will want to perform it is convenient to restrict outselves once and for all to considering only unknown functions yx that have continuous second derivatives and satisfy the given boundary conditions yx y and yx2 y2 Functions of this kind will be called admissible We can imagine a competition which only admissible functions are allowed to enter and the problem is to select from this family the function or functions that yield the smallest value for I In spite of these remarks we will not be seriously concerned with issues of mathematical rigor Our point of View is deliberately naive and our sole purpose is to reach the interesting applications as quickly and simply as possible The reader who wishes to explore the very extensive theory of the subject can readily do so in the systematic treatises 66 EULER S DIFFERENTIAL EQUATION FOR AN EXTREMAL Assuming that there exists an admissible function yx that minimizes the integral 1 fxyy dx 1 how do we nd this function We shall obtain a differential equation for I See for example I M Gelfand and S V Fomin Calculus of Variations PrenticeHall Englewood Cliffs NJ 1963 G M Ewing Calculus of Variations with Applications Norton New York 1969 or C Carath odory Calculus of Variations and Partial Differential Equations of the First Order Part II Calculus of Variations Holden Day San Francisco 1967 i nieeeeeeer 1eL E2IJ lM TIEH5 yx by eemparineg the VEJIELEE Df J that eemereespernid to neighhnerienge eedmiseyiielee funmines eeentral idea is that since yr gives a mimmum value In L I will iinereeee we deieset11e1rbquot yx sligeh iy These diiS11UIIquotb d funquotetien52 are eeneteruvetede as fellilewe Let are be any fuinctien with the preepertiees thal w x E ti3FI39L1Er1l5 and em 5 E2 02e 3 If e is a Sme eperemetetrz then NI 2 WE MME L3 rsepresents e emeeepereemeteer femil3 ef admiesihle mneetiens Thee vertieeaxl dr EV Ei n ef ee e39urve in family fmme the meieeniameizicng curve yr is equotq I as eheeewn in Fig The eini een ee of 3 lies in the feet that far eeeh ffeimily of this type that for each ehubiee ef the feumeteinn rg1e the eminimieinVg funeitien 39x 39belenge In the family anui mrrespends In the ve ueree evfV the peran1eter cr N ew e weith nrx we seeubstietuete j3 Fyr xj m3939e and FWGUEE M lThe diEerenee A y an is eeel ed the eerfeer ene ef the efueeiieen and iie ueueiilegve denrgnned by y wee Iien een dee39eIeped ifI39 D 3 ueefutli feerme4 iem which we dc net dii5e39uesj and is the eerer39e ext the name c39e Eu eu f ee1r39 aI39 iere THE CALCULUS or VARIATIONS 507 7 x y x m7 x into the integral 1 and get a function of a 1a x2rltxyy39gt dx fZrxyltxgt cmltxy39ltxgt an39x1dx 4 When cr O formula 3 yields 7x yx and since yx minimizes the integral we know that 1a must have a minimum when a 0 By elementary calculus a necessary condition for this is the vanishing of the derivative 1 a when a O I O O The derivative l39a can be computed by differentiating 4 under the integral sign that is ma x rltxyy39 dx 5 I By the chain rule for differentiating functions of several variables we have 8 8 8 8 839 8 8quot fx 39 7 P P2 P 8a 8x 8a 8y 8a 8y 8a 3f 3f H r ayI 7135 so 5 can be written as 2 8f 8f 139 2 J W 1 d oz M ay nltxgt 3 n x x 6 Now I O 0 so putting cr 0 in 6 yields 2 3f 3f d O 7 Laynxaynx x t In this equation the derivative n39x appears along with the function nx We can eliminate 1739x by integrating the second term by parts which gives 2 Bf n xdx nmgy ij fang 5 x By d lt 81 I d L nxdx ay 6 by virtue of 2 We can therefore write 7 in the form 2 8 d 8 1 vltxgtli gtdx0 z x Y 8y dx mFFsiaisi39r maa aanuasisas Dar raas niitng U in this paint is based an a ssci sl mis a sf tiis fiunsisiimi Haiiwraaari siiinaa the intagral in 8 must vaniish ins awry such fimstiian we at miss canclud a that me axpr asis ian in ibra satiis must salsa vaniis h Thisi jgiaids J T 3 3 p I it as 3 wisish is Euia s EiqE39EJm i I39 V f E l II is imp3rta nt its have a clear uniclisrstanding the mast nature of mglr ssns iusisn namsiiyii yi is an admissiibis funaiinn that minimiaasi imegrai 1i th v y Eauler s aquiatiiani Suppasss an aimissiiibla fsiinmiani y can his fl1m ti1 that S iiii TFIELS thiis squatiari sass ii1is mass that imiinimiass Nat insirziassairiiy The s ituaIiun p similar in that in aiamamary ciaisuiiuis whaire is funciimi glfx wimsss dairiiviatijvs is zero at a stint an may NEVE a zr asimuimi a minimum ass a psiim sf Eii ti i at 3quot arm diistiis1 rrtisns are made thesis cases are sftain saliad siarisnary Lraiaas sf L and tha poims is at which tiii ijsi Data are s arisnaryi i3 i2ifSu In the sama way tiha caindTi39iisa iquot sari psrif aatly wall iridiiicate a rnaaimum Cur psixrii of i ECITii3I1 far Ha at W D instsaid sf a mimiimuimi Thus it sustamiasy to sail any admissibia SDiUIIi o Euier s aqiuati0n a siaisiaaary fimaIiaa sari Ef rli ry auri1iai iaimzi is rafssr is the csrraispandiing viailus of this integral 1I as a sraiikanary 1i 1fuE sf this intsgrai wiiIhsuti c mmiittiag rUI1iIiSEi39VES as is Whiil h of the siavariai p SSiibiiIiES E39t1i iiI39 ascuirsii Fnmharmsrrei sa luiisns sf Euisss EqU Ii H swhcishi are unrsstriciad by this ibaunzdary sanidiitiians ass saliiiadi axrramais In ca1wuilus we usa this S v nd darivatiivs is give sui ciani I3D139iiiIiw 3nS disiimguishiinigi ans typa of st atim1ary vaiiua imam anmihar Similar su aiazati E I391diTIi39IDrl39i5 airs mraiiaiibia in P icailciuhis sfvariaisti0n1s suit sinna fhissa are quite campiisatadj we will not canisidar them harem in actual priaicitics the gsam atr ctr N sf tins p139n3l em uninzlar discussiimi siftan m i39iES it psssibie to datasrmins whether a p ftiC Ui l1quotSIiE3iiD lquote39 fumIim1 maximizes or miniimisas itlhs sini tiegral rr nsithar This fc39 iET39 who is miarsststi in s39uaisnt icansiitiiiicns and Ei39li39iEiIquot thasirsticaii pmbilamsi will find a dsquata dissuisisiianis in the masks man5ti tmasi in Sactiiam L stamisiii Eguiia s aquaitiiasn 399 is inst vary illuminating In asdar is Lil39l39El39pFEI1 it and minivsrst it iriia a Llis fliil armii was bagiin by amphasiaiing Jim mars dE t I I tha iridiiiraai iargumam isadin in E9 is as fD n139i3li quot5ia Assuma that the braallzaia fmir39li a in E nas saw isay p asiiiaa am sama puzriialz 1 a in the in araaL Siinsza this l39iunsisim is asnxtinuausi it will has assiliss iihira ughiauii same 539Ubi 13Elquotu Ei aizwaaut a 5 a Charass aufn nsr that is ptasiiiaa insids ths s39uhinlarvai and saris cuitsiaa Far ti1is srmsi aha integral in 3 1a ili bc sasi tiisauwhicLh is a DDHiI1quot3iiiJ39EliD s Wham this iairguaiianft is f fin ilii d tiha rasulEing siiaianiaiai lmuiiwni as aha fans a39maaIIai39 ismrim sf was aaiaraias sf uaraiaas THE CALCULUS OF VARIATIONS 509 that the partial derivatives 8f8y and Bf8y are computed by treating x y and y as independent variables In general however 8f8y is a function of x explicitly and also implicitly through y and y so the rst term in 9 can be written in the expanded form 3 ii LdYlt f4Y39 5 8y 8y 8y dx 8y 8y dx39 Accordingly Euler s equation is dzy dy fy y39Zquot fy yEr fv x 39quot v 0 10 This equation is of the second order unless f 0 so in general the extremals its solutions constitute a twoparameter family of curves and among these the stationary functions are those in which the two parameters arequot chosen to t the given boundary conditions A second order nonlinear equation like 10 is usually impossible to solve but fortunately many applications lead to special cases that can be solved CASE A If x and y are missing from the function f then Euler s equation reduces to dzy fry 55 2 0 and iffy 0 we have dzydxz 0 and y clx C2 so the extremals are all straight lines CASE B If y is missing from the function f then Euler s equation becomes d it 2 0 dx 8y and this can be integrated at once to yield the first order equation 3f C By 1 for the extremals I CASE C If x is missing from the function f then Euler s equation can be integrated to Sf ay fquotC1 51 nw aaamat aauaTmMa This allawa tram the idantiity at at z 7c at an at Cm H J E a i a 4 P ah By a air Jay 3y 8 aintta Sf at ii U and itha axjpraaaiuat1 in brackets an the right is aata by iEaia1quota aquatitmt We naw afpgpiy this mtaahinary tn the HIIBE ptabiama fnrtmuiattad in SECtti P Ezmmpla p To nd the shortest cmaa jaining twa paints iaMy3 and ix391y w1htiCh wa knaw ittturiittivaly to be a attraigtnt tina wa must minimize the art iatagth integral TI39Tt E taquotaIquot iablES st and j are missing tram ft nrab Eam falla tmdar Caaa t Sitttgza f M V I R Casa A rta a as that tlrta aafttamafla L39atquot the twa paaramatar fatmily at straigh tinas ya 2 eta M That Wundatj aa ndittiana yiaid pN i an this a5 H Fa M M at at as the a tatim1attquot at1ra a and this is if Cauraa the straight iitta joining thi tw pai1tta It Si39ttJM ba nmad that this analyais ahuwa mini Iirhat if I has atatmiary utatxaa than th a CiUITE5PU lji g ataitianary auirtra must be ti atraight lina 11 Hlawiaaatr H is Claar 39fn39Jm quotthat gE 39tT1E ElquotjF that E has zr maaiimiaing catrva but draaa hava a minitmiaiing tauntat an aha ctmcludia in th way that 113 act39aatllja ia the aiimrtaat auwat jiaiitning mar two parinta In this aatamtpla we arrivtatdi at an ahviavua caaaiuaimt by aIquot39t iytiE means A much mate di cutt aintd iI1tEI ESting pmblam is that at ttdii the atnartaat aiuraa jaitaiquotag two ztadi painata at a given aarfaaa arm mt etnttitfaly an that aurfaaa Thaaa auwaa are aailad gaaidaaiitaa and the 5tli1 atfi thair jptarpattiiaa ia ma at this f l of that iiir39 I39liEif at mathamati kniwn aa di aratttim gaatimtatty Example p Ta fintzi tha autrvaa ivaittittg tiha pittittftas fxhyli and iI33yg ti yialda a aurifaaaa G39ft E eUiL1tit397J at mittimum area when raavnltvad aabaait the atria we must mtinimiaa J E Ji 1quot Ca it THE CALCULUS or VARIATIONS 511 The variable x is missing from fyy 2ny1 y 2 so Case C tells us that Euler s equation becomes yy392 P h WW y1y CI which simpli es to em Vyquot ci On separating variables and integrating we get C J C0gX VY239CiC l 1 cl 2 and solving for y gives y C1 cosh X C2 13 CI The extremals are therefore catenaries and the required minimal surface if it exists must be obtained by revolving a catenary The next problem is that of seeing whether the parameters c and C2 can indeed be chosen so that the curve 13 joins the points xy and x3y2 The choosing of these parameters turns out to be curiously compli cated If the curve 13 is made to pass through the first point xy then one parameter is left free Two members of this oneparameter family are shown in Fig 95 It can be proved that all such curves are tangent to the eltIgt I I I I I I I I I I I I I I I I I I I I 1 XI gtHI FIGURE 95 512 mFFEREHTw EIUUATJTUHE dashed curve SH 110 cunE in the famil1mr39m55amp5 Thl sj when Eh 5 ecgtUm pnint It2y2 is Abel w as in Fig 95 them is Il catenaw thr0vL1gh bow puims and n smativnar1 mnction exism In this case it f fmlnd that smal1e and 5ma iEer su39rf39ace5 are gezrlegrzated by Acuwes tjhat approarsh the dashaed Him fmrn tlty tr Kxhl ji to t1I to rx1y1 SE1 mm a dmis5ib IE cum mi g enera1e 3 minimal 5urfac e Wham the semnd Vint lies abmre C there an two cazttnarics thmugh PX pointT and hence tum 5I IIiJ I I l39jquot39 functiUna 39lriJ only the upper caIe1nary generates a minimal surfacc Finalh when tha sccV0ndl paint is an C there m 1lly ane statziUnary functi n but the surffa e i gengrates is 17B D T minimal quot l ixamplle S Tn nd the u rv e uf quiampcke51 x5a4ent in Fig 93 WE mus mmimize cP red39IIc5 I0 JJU V39 E whicll pXS precisly the sIii 7Em1tTial equa 1iIrn 639 1i arrived at in u earliea di5cusvsi n f this fam t3Vus p mbhm its Nuti m sis gimsn in SEctzia n e Tm rresulting static1nar39V curvE is the cycluid Jr E vnlf sin 9 and y M1 E 1295 6 M3 gengrated quotI33 3 ciracfmrE Elf ra dius uquot mIIing wander the 1 ai5 when 2 is ChiquotEIEIT 3D than Eh rs iTl VEfIEZi armzh passcs thmugh the paint Ax2y39g in 2A 3991 A bef m i argum nt shows nm y that if I has a minimAumi than the cnnespondjng sVtJaI39innary azurwz mu5t be the E 39EW7 id Id Hwuver in i5 wasrJn1a39bVy clear frmm Ph39L5iC i cinn5idemtiun5 than 1quot has no maximizing scum but d es haw a m39ini mizing xcuwe 51 1his c339clmi4d HENiI aI1Ey minimize5 tThe E39i39EI1t 0 dfi5EEn1 We Lznclude se t irmj with an Easy but impc3 rtam ejwcimnsi n mi mu treattm nt mi thus integral L This integml rep1 ESE4nts vai ami naJ pmb erI1s Cuf tht simpra5I Wp because it inwJIv es Dnly LIHE uri1knuwn functinni II wrei39ampri mime f the svituatinns WE will enc unIEr bebll w arc n0t quite so simpflia far they lea In iT1TEgIquot l5 dc psmdlng min two gr l 7I 1 f39E Lmkrmwn f39unctim1s 4 A EIJH i5 cUu55j n Gf I39hE395E39 5 fat mE1IE3 wirih prufs can be muxndl izn Ci ha pTEI W Uif Eli5 5 C C 39E39E39uti39m VariariEJIFLH CELTL15 MUnusgzraph mm 1 Mathamprm lfIa a 5t5Eli IiIUamp D1f Ar1ri naL 1925 THE CALCULUS or VARIATIONS 513 For example suppose we want to nd conditions necessarily satis ed by two functions yx and zx that give a stationary value to the integral I zf fxyzy z dx 15 where the boundary values yx1 zx1 and yx2 zx2 are speci ed in advance Just as before we introduce functions nx and n2x that have continuous second derivatives and vanish at the endpoints From these we form the neighboring functions 7x yx arnx and Ex zx m72x and then consider the function of or de ned by 1a I fxy crnz an2y cr17z It crng dx 16 Again if yx and zx are stationary functions we must have 1 0 0 so by computing the derivative of 16 and putting a 0 we get f2e3 er 8 er aym 9 772 777i 5z 7l2dx 0 or if the terms involving 7 and 71 are integrated by parts lfilnultxgtl 4ln2ltxgtl ildx0 K Finally since 17 must hold for all choices of the functions nx and n2x we are led at once to Euler s equations 4 if 9i 2 51 if dx 8y 8y O and dx 82 82 039 18 Thus to nd the extremals of our problem we must solve the system 18 Needless to say a system of intractable equations is harder to solve than only one but if 18 can be solved then the stationary functions are determined by tting the resulting solutions to the given boundary conditions Similar considerations apply without any essential change to integrals like 15 which involve more than two unknown functions I PROBLEMS 1 Find the extremals for the integral 391 if the integrand is V1 y 2 3 y b y2 y 2 V q39 y r at the tindctpend B ntt variabltt 39CnSidEt39 twtt p it t39t 5 F antil Q an p 5 r f 39EE mt th sgpthtAte 11 y 21 a1 and tamrdi ina1tize this EUt39faCE by mte an5 of the 39pIr1etita tt2ttJfJiittta39EE5 tamzlt FALL ECUt5t TtUME E Find tt1te sttatttzvnaw zl t ffti n f W ax tvhictt ts dettmmtntett that t39 U 7 fF ctwindiititonst yt t U and yt1t3 P quotWhen tyre tntezgmttd in tt if the farm tIJtJ t E KE Ty2 F slmw Eh 39l E39uterquot395 eq39LtatitIn ta ta semnd tttder tinear ditt ttatr tii t tequatiun P an Q are tutu paints in at ptattc than p rtterrrts of pnrtar E2xD39IEtft iit taquotquott thttet length of H cunret fmm P tnA Q is Q Q M t I I tfdtrg rld z an p F Find the putat eqItattntt Df a straight line by mtniImi3tttg this i tEgI 3W aj wi t39h 9 at the itndtepentdantt varitabta E Wham I 39 H sirrt vwu5 H 3 1 sin ER and z at CH5 Lt1 H 2 Fm be a tcuwet t ng an the zsurfacae antd j inttngt P and q SZh t3wt E39I E the 5hm39tte5t smzh 39urvet at gtndt3ti is art are Df 31 gIquott3tt ctirctes tthatt is that it lies tm a pl tt f tthrtmgh the vCEn tEE P 39Eft39pl39t St5 the tcrtgtt1 of ttm tturwe in the f39DT39I39t t 9 tr T A Jt A J 4 ayt dz wp F tt39J I p shirt gtvdqftj 50hr E th usrt ttsp rt ttttng Etttr rr ettquatjtun fmr Hi and umt art the ft E5JLttI back intt retttangt1ttItar c rdjt l t It 39 I H Prime tttatt any ge desic M the tight circular trum E1 t t N 3 E 1 that the tultwing pt Dt te F39E39t If the cane tit along 3 g Eit39a39tUI Emil I i rt intim as 1 J393t tEv tihan that gE dE5 i5E b fjm 3 tst ttratght ttinat Hthtttt ttept sent tttt C i fJ f tquot iTiE quot by tmeatts tuft th Eqt1a3 tifJn5 ti 5 t l E37 1 H J 5hm wt that the parametert r tand p TIBpI39E5EH1 Uztdiinttryr jprutar EU39Gtt39EtT7l I39E5 nan 39 at39EItBd ctttteg attnd tlttzuw that 1 g39EftCt1E5tI r 09 rtt t a straight inE39 in these pntat tc31wt1tditna tet5 If tht ctutrw 39 gtzju is 139E39t397tJHr39Eid tatmtut ttttt E F39tttt5 ttten tilwe f39EjiS LI2I Iit tg surquotfaue 1 mtrttlutitm has tquot y g39z2 as its equtatttun A anntrEt1ient patamtttt re39prt3ettta ttnn tjf this Stlttf CE ts Agriawn P f t gz 1205 St g sit 6 I THE CALCULUS or VARIATIONS 515 where 9 is the polar angle in the xy pane Show that a geodesic 9 92 on this surface has c2 V1 g39zquot 6 quot gzgzgt2 ct as its equation 8 If the surface of revolution in Problem 7 is a right circular cylinder show that every geodesic of the form 6 62 is a helix or a generator 67 ISOPERIMETRIC PROBLEMS The ancient Greeks proposed the problem of nding the closed plane curve of given length that encloses the largest area They called this the isoperimetric problem and were able to show in a more or less rigorous manner that the obvious answer a circle is correct5 If the curve is expressed parametrically by x x39ti and y yt and is traversed once counterclockwise as t increases from t to 2 then the enclosed area is known to be 1 Z dy dx quotET 2 lt1 which is an integral depending on two unknown functions6 Since the length of the curve is T 2 dx 2 2 L f H H dt dt I 2 the problem is to maximize 1 subject to the side condition that 2 must have a constant value The term isoperimetric problem is usually extended to include the general case of nding extremals for one integral subject to any constraint requiring a second integral to take on a prescribed value We will also consider nite side conditions which do not involve integrals or derivatives For example if GxyZ 0 3 is a given surface then a curve on this surface is determined parametri cally by three functions x xt y yt and z zt that satisfy equation 3 and the problem of nding geodesics amounts to the S 5 See B L van der Waerden Science Awakening pp 268269 Oxford University Press London 1961 also G Polya Induction and Analogy in Mathematics Chapter 10 Princeton University Press Princeton NJ 1954 6 Formula 1 is a special case of Green s theorem Also see Problem 1 DIEFEHEquotH TIAiL 39Evl39ZNJATEJ1MS pmbliem U1 minimizinAg tha are icngth integral subject 9 the side canditinn 3A Lagrange muItipliEr It 0 IquotiECE5S l39F In begin by c nsidering ssme pr hlems in liemEn39Eary calcuius that are quite similar to is perimat39ric pmblEms For example suppuse wc warut E6 nd the paints ry that jiEid 5tatitmarv vames Lfnr 3 fll i 2 ry whEre Vhuwever the wmariablcs J5 and B are mt ind pendenAt but are constrained by 3 sidla cnwn minn The usual prncaediure is In am39btrari1y dESignVaIa mane f the variables r and Q in 5 as ndependentV say I and the other as Adependsent m1 it an that dyf can be c4umpuI e d frmfn amp I 51 Eydx EVE nmt use the fact that simzte 2 is now a mmmn of 1 almaac afzfia D is 3 n ece5sary ctaaAn diti UrVn fm 2 to have 3 stati mary yams 50 d 8 nix 51 m E 0 at 3 8g395 y p Um sU lving 5 and SiT UIEHEUUSIE we obtain the Vrcquiredi Vpnints IJAA HE drarwbak tn thi5 Eppl39WI39 Ch is that the variabIesV 1 and crcc4uI sy4mm etricaIly but are treated umymmetricaily E1 is pnssibler 10 Somme the same pf biB m by a EifiErE n39 and mm e4lega4nt miethmd that also has many practitsahl advtantage5 We fmnm Ih f unct39aEGn FAI M 2 ltm I rm and inw3sIigat 3c 39 HE v 3fF in d taIirnna1ry va lLues by 1116 l7I39S f the 3963 In very airmplc CESES1 of wursc we can 5nhm 5 fair 39 as 3 f39um1im nJfT 1nd irmtart this ir 3 f 39I whi th gi ve 2 as an mpEiit fliE 39 39l m39J mlquot Jr and all that rernmns L5 In JmpMu drfidx salve the equation zflir H and nd thug cnrra pnixmding THE CALCULUS OF VARIATIONS necessary conditions aF a f 22 0 8x 8x 82 erquot a a 1 A5 0 7 8339 8y 8y aF 0 31 gx y If 1 is eliminated from the rst two of these equations then the system clearly reduces to 6ic9j 8gBx 0 8x 8y8gBy and this is the system obtained in the above paragraph It should be observed that this technique solving the system 7 for x and y solves the given problem in a way that has two major features important for theoretical work it does not disturb the symmetry of the problem by making an arbitrary choice of the independent variable and it removes the side condition at the small expense of introducing 1 as another variable The parameter A is called a Lagrange multiplier and this method is known as the method of Lagrange multipliers8 This discussion extends in an obvious manner to problems involving functions of more than two variables with several side conditions and sxy 0 Integral side conditions Here we want to nd the differential equation that must be satis ed by a function yx that gives a stationary value to the integral 1 r2fxyy dx 8 where y is subject to the side condition I2 J I gltxyy gt dx c 9 and assumes prescribed values yx1 y and yx2 y2 at the end points As before we assume that yx is the actual stationary function and disturb it slightly to nd the desired analytic condition However this problem cannot be attacked by our earlier method of considering neighboring functions of the form 7x yx crnx for in general 8 A brief account of Lagrange is given in Appendix A rJI EFEiiTl lii E0U A5 FIDHlt tha5 e will not maintain the smmxd integraV J at me C f t inl vzame 2 Instead we consider a twr3pararmeter famnfmily oil neighboring f um39Iim1s 2 p 0 P quotwhare hxV and n 3r have mnJ1im11mu5 sezmnd d iquoti quot t39ivBS and quotvan i5h at the EmiippDimVt5 The pamparamkEt wrs arm an Q are nut indE39PE dB E but are rcalamdi by the cnnditi n that IE T N Our pm 3l m is r7th En gred umd to that if n i5iing neVce55ary mn ditinn5 for the fuxnctiun MmmwfWmwwr m EU have 3 5taIi 1nar 5r VEUE at tr z RT 4V w hart m 5aI i5fja39 A141 S ll illiiilll is madg 10 order fr the methsd af Lagrange multipliers we therefnre intrndur E the tfunctmn I M wilmre I I f and invesIigatAe its uncnnstrai4nEd 5Iatinnar394y39 value at at 3 E by means at the 39mE EB S FF cmnditifir if E Bur E 612 P If we di3Eti39 Er ntiat 3 13 under th inmegraI P nd use 1 lquot3I we get 1 A p SF J 39 A M by MI SE 71 I1 I an and settrijng mm IT J1wi6ld5 1 G i 6F V F wmmm m U ID whm em mg p M by virme of M After the sEc0nrl term is int6graI adV by part s this hEJED39mES 3 3912 A SF L 6355 j a A 151 Vzf Ur sf 15 A q xL3 y it 4A3W I 39 Si mrir and nEr are both arbiIrary the two mnmtiii ns E 1JUDdiEd in IL5 amcmnt 10 only one mindAiti 0n and as usual we mnclude that the THE CALCULUS or VARIATIONS 519 stationary function yx must satisfy Euler s equation d 817 8F dx 8y 8y The solutions of this equation the extremals of our problem involve three undetermined parameters two constants of integration and the Lagrange multiplier A The stationary function is then selected from these extremals by imposing the two boundary conditions and giving the integral J its prescribed value c In the case of integrals that depend on two or more functions this result can be extended in the same way as in the previous section For example if 0 16 I f2fxyzy39z39dx has a stationary value subject to the side condition J I gxyzy z dx c 139 then the stationary functions yx and zx must satisfy the system of equa ons c lt9F c 9F0 an 44 c3lFO 17 dx 8y 8y dx 8239 82 T 39 where F f Ag The reasoning is similar to that already given and we omit the details Example 1 We shall nd the curve of xed length L that joins the points 00 and 10 lies above the xaxis and encloses the maximum area between itself and the xaxis This is a restricted version of the original isoperimetric problem in which part of the curve surrounding the area to be maximized is required to be a line segment of length 1 Our problem is to maximize I 393 y dx subject to the side condition I J V1 y 2dx L 0 and the boundar conditions y0 O and y1 0 Here we have F y W1 y 2 so Euler s equation is p 1 0 18 or after carrying out the differentiation yr 1 2 19 1y232 A 5E E GUMJ39Tl E 5 In this tag mI int agrati n is nece55ary siince 19 tells us at fl that the uurvature is constant and equa1s MA It fm wzs that the r squired i maximizf ing curve is an arc Hf a eir r e as might have 39biampen expected with radius P7 an alternate p rme39ure4 wa can imcgra te I13 120 36 F g quot 0 13 Ay 391397g 1 On s lving fur yquot md im gra39ting again we Ubt inl I 139 Ea U E24 33 which Elf murse E5 lhc equatian f a circle with Iiadiug 1 my Efx mpale In Emmm3 391 it VIeariiy nccessa4rj m have L 2 L Aim F L 33 M2 the circuVkaI39 arc determinued by Em will rmt dez ne y 2 B as 3 singIeevaIu td funmion f 1 We can amid these arti cEa I imuaa by uansiVdiering wwes in parammric EnVrm Ax 2 mm and y Mr an by turininyzg our at1eVn1iun In the uriginaI i pEfimElTiC prahlem Elf maxquotimizing y 2 I M H w hare 1 d a39r any rdyh139r wi1 h the side umndi Iinn 391 Hare we has I I F P y y1 p5 ii il hi an the Euler eqfuatimsns W arc and ia lj 439 Egg D K If we S DhFE far 1 D 311 ch squarelt and asi l thwen pK resuit is 3952 Em 2 quot12 50 the maimizing curve z a wziircirc 0 rcsuit can be expar ssed in o fCi1NUW39ilE Ig wayz L its 11 13ngth of 3 clursed plane curve that enrzlmea an area JL Atlhsn u E L14JT wiih ampqua1ji I3I if and I1 quot if tht curve is a irJiE A relea39tim1 rsrf this kind is Aca cd an E5Lmperi m rri lEI E il Eilfiiliij aq quotquotSIudentsa J f pl i39Sizl395 mew he in1IEi 39E51Eltd in the idra5 discLu 5ed iim P m PtI7ya and p h 5fEgrL39L I 3390pErf39me rf nequ a39iirr39e in M39mh39 mu 1lt cm P yji Fquotri39ncttan Llniv139sit3r Pr353 PZ39i EEiDni f p d THE CALCULUS or VARMTIONS 521 Finite side conditions At the beginning of this section we formulated the problem of nding geodesics on a given surface Gxyz 0 21 We now consider the slightly more general problem of nding a space curve x xt y yt 2 zt that gives a stationary value to an integral of the form Ffiz39 dr 22 where the curve is required to lie on the surface 21 Our strategy is to eliminate the side condition 21 and to do this we proceed as follows There is no loss of generality in assuming that the curve lies on a part of the surface where G2 a 0 On this part of the surface we can solve 21 for 2 which gives 2 gx y and a a 25 x 5y 23 When 23 is inserted in 22 our problem is reduced to that of nding unconstrained stationary functions for the integral 392 8 8 In fxy ix 8 at We know from the previous section that the Euler equations W6618 for this problem are f Jf I5EE9I 0 dt 8X 8239 8x 8239 8x and i2I 2J E0 dz 8 8239 8y 82 8y It follows from 23 that f9Z and ZZ 8x dt 8 8y dt 8y so the Euler equations can be written in the form d3 d J39 and i ltEd I0 Z1 ax 8x dt 8239 dt ay 8y dt 32 If we now de ne a function lt by d 8 A G 24 dt 82 r 2 lt gt 522 ID Ii39FEREHFH 39iI EQUATIONS and use the I1 EiLi3LIi I1S Sgf x i j and agfay j G33ir39fE than Euie s equ timin5 bec mei d fj e 39 VV c V A PL D dr a in I t J and Ar39Gy 26 Thus a nece5sary cundiii39tLi0ini far a Stiaii inairy miuc is the E lii t of a fuilctim391 1I 5ati5fyinig equati ns 25 and 26 Uni Eiimiilrmting 34 iwe Dibitaiin Ih symmietriic Eiquaitiiains idxdiafai F dfd i8fif3 M dxdiafxaz39 G 39 G quot G Z IFquot which mgetheer Wlkitih 21 deterrriine the exquottrernaIs Hi the pi biEm It is iwmth ramarkini that equiaiIii miris 24 25 an 26 Eian be regia ridied as the Eiu ler E q39M ti039 5 Aim thra prnblem Di miiing unc nstraiincd stati0nairy fuinctimins for the iintegrai 2 H I PN v an i This is veiiriy similar It our mnciusainm far integral Aside E39D iilii I13 EK E39EP ii ithsat hJarti the miulftiplier is an undterminedi function mi I instead iii an unieitErm in edi imiinistant Wham we pE iHiiZE this rasuit v tlha pmbliem mi ndiing g emdeisiiics an the surface 21 we have This equalziuna 2 bwsc mci ii iii iiiiwiif R iidmiiiiifi Wdififiif P J and the pirioibmm is to extrac39t inf rmatiGM frmm this syS39iem Example 0 If WE change the suiifate E1 rm Ah the E phEiquotE xi 12 ith n Cixyiz39 Ail 1 ya 539quoti 31 and 23 is q i xmi z ihkfi which Can be rEwriME1iA iiri the farm If C S U N p w w fyb THE CALCULUS OF VARIATIONS If we ignore the middle term this is ddtxy39 yr ddty2 2 Icyquot yi yi 2 39 One integration gives xy yx cyz39 2 or xc2y xcz y and a second yields x cz cz y This is the equation of a plane through the origin so the geodesics on a sphere are arcs of great circles A different method of arriving at this conclusion is given in Problem 665 In this example we were able to solve equations 28 quite easily but in general this task is extremely difficult The main signi cance of these equations lies in their connection with the following very important result in mathematical physics if a particle glides along a surface free from the action of any external force then its path is a geodesic We shall prove this dynamical theorem in Appendix B For the purpose of this argument it will be convenient to assume that the parameter I is the arc length 5 measured along the curve so that f 1 and equations 28 become dzxdsz dzydsz dzzdsz 29 G G G PROBLEMS 39 1 Convince yourself of the validity of formula 1 for a closed convex curve like that shown in Fig 96 Hint What is the geometric meaning of Q P JydxJ ydx Q P where the rst integral is taken from right to left along the upper part of the curve and the second from left to right along the lower part 2 Verify formula 1 for the circle whose parametric equations are x a cost 66gt and y a sin I 0 S t 5 Zn 3 Solve the following problems by the method of Lagrange multipliers quotVa Find the point on the plane ax by cz d that is nearest the origin Hint Minimize w x2 yz 22 with the side condition ax by C2 d 0 b Show that the triangle with greatest area A for a given perimeter is equilateral Hint lfx y and z are the sides then A ss xs ys z where s x y z2 c If the sum of n positive numbers x x2 x has a xed value 3 prove that their product xx2x has 3quotnquot as its maximum value and THE CALCULUS or VARIATIONS 525 ics He also contributed to number theory and algebra and fed the stream of thought that later nourished Gauss and Abel His mathematical career can be viewed as a natural extension of the work of his older and greater contemporary Euler which in many respects he carried forward and re ned Lagrange was born in Turin of mixed French Italian ancestry As a boy his tastes were more classical than scienti c but his interest in mathematics was kindled while he was still in school by reading a paper by Edmund Halley on the uses of algebra in optics He then began a course of independent study and progressed so rapidly that at the age of nineteen he was appointed professor of mathematics at the Royal Artillery School in Turin Lagrange s contributions to the calculus of variations were among his earliest and most important works In 1755 he communicated to Euler his method of multipliers for solving isoperimetric problems These problems had baf ed Euler for years since they lay beyond the reach of his own semigeometrical techniques Euler was immediately able to answer many questions he had long contemplated but he replied to Lagrange with admirable kindness and generosity and withheld his own work from publication so as not to deprive you of any part of the glory which is your due Lagrange continued working for a number of years on his analytic version of the calculus of variations and both he and Euler applied it to many new types of problems especially in mechanics In 1766 when Euler left Berlin for St Petersburg he suggested to Frederick the Great that Lagrange be invited to take his place Lagrange accepted and lived in Berlin for 20 years until Frederick s death in 1786 During this period he worked extensively in algebra and number theory and wrote his masterpiece the treatise M canique Analytique 1788 in which he uni ed general mechanics and made of it as Hamilton later said a kind of scienti c poem Among the enduring legacies of this work are Lagrange s equations of motion generalized coordinates and the concept of potential energy which are all discussed in Appendix Bquot Men of science found the atmosphere of the Prussian court rather uncongenial after the death of Frederick so Lagrange accepted an invitation from Louis XVI to move to Paris where he was given O See George Sarton s valuable essay Lagrange s Personality Proc Am Phil Soc vol 88 pp 457496 1944 For some interesting views on Lagrangian mechanics and many other subjects see S Bochner The Role of Mathematics in the Rise of Science pp 199207 Princeton University Press Princeton NJ 1966 6W etFFEatssr39rLaL Et3tiaT1tst s apartrnertts in the Letaare Lagrtahge was leatremeljy mtdfetst ahtl manag rhatic fer a mean at his grseat tarlcl th e ugh he was a friet tEi at alristecratts uar1ci incised an taristecrat hlmself he was respected 39l tti held in a quot ectiert by all parties tchrieluggheult the illt lTl ill eat the French Revelrul tlIris mast ihtpertattt writ dtlring these years was llieadilhg part in etstttabiishihg the tttetric sysstieml elf weights and measures In mathematics he triecl te pretrtidfe a statistfactetry fetmtlatlen fer the l1asic precesses hf ahalysis but these re lsrts were largely aheirtit39e Toward the end at 0 lite laagralngle felt that mlartthernatilucs had reached a diesel ertrsl and that ichemistry pihysilics hieifletgy and ether SEiEl lEES weuld attract the alzllest minds at the future His might haquotire heart retietredi if he had been ahlle te tertsee the cethlrtg est Gauss hand his successars Wh made the nihe tecrlth clentutry the richest in the lung hiisttery cf mathemafticst ITS lMPLllCATlUNS One purpelse ef the liT39llti3liill E37IilEiii i of the eighteerlth clentury39 was tel disrcewetlr a general principles trclrrl whichl lhlcwtelhiianl li iiili titfitt ceuld hee dledittcedt ln searching fer clues they hated at nhmberr sf cttrieas facts in elemerltary pfhlysics fer example that a ray elf lightt fellears the tqutlclcest path threttgh art laptical i t5lECli l1l tIil that the ietquiflihriutttl shape at a hahgihg chain rfnirtirttiaies its paterttialil EC gtEigquot and that seaip ib lli3tblvES assume a shape hatrinlg the least surface area fer a given velun1 e These facts and ethers slaggestecl te Euler that natttre pursues its cllsiversle erlcls lay the tttszt et jcieat land ecenemi cal imeahs and that hidden simplicities anideirliiite the apparent chaes ell pihenetttelna it was this metasph3lsilcal idea that led him tel create the callctlilius at vtalrsiatxlelnits as a teal lier ilhvesttigaltinlg such questiiehtst Esuleris dream was rerallaecl aIrnaslt a century clatter dd ilclarrtilten Iiamilstnis lI iIttiple Censivder a particle at mass at teasing thraugh space llI39l39 dEquotT the in uence at a farce 0 d o f g and assume that this telce 0 e cstalssrasarilare in the sense that the werlt it d39U E2 in I39l t39Elquotla39i39tTlg the particle f39li llt erte pveirlgt the attether K lli39ildI 3ipE vdE39 I all the pastime It is etasy ate Ei39lUW that 1 here EIEi5IIS a scalar iaihctien Uxtia suclct that area 4 F 8Ul 539p T and lll la E X The f ullti t39lit 2tl l zV Q E wN is called the Jsteatisl enlargy ef the partlclei since the rcharlgse in H la the la rtatJage etf ae cte39t aaatl3tsis F is the gzraditentt ref U y alt dzdt and if the action has a stationary value then Euler s equations must be satis ed These equations are dzx 8V dzy 8V dzz 8V 0 0 0 mdtz 3x mdtz By m dtz 82 and can be written in the form dz 3V 3V 8V m n 1 dtz Bx By This is precisely Newton s second law of motion Thus Newton s law is a necessary condition for the action of the particle to have a stationary value Since Newton s law governs the motion of the particle we have the following conclusion Hamilton s principle If a particle moves from a point P to a point P2 in a time interval t S t lt t2 then the actual path it follows is one for which the action assumes a stationary value It is quite easy to give simple examples in which the actual path of a particle maximizes the action However if the time interval is sufficiently mFFEEiENTZiAL auuameea ehert then it can be ShDWW that the aetien is n eacneaaaril3J a miniimume In ihis ferm Hamiltteawa prianeipelea is emeatimee ealled the pnL ne1fpEe ef fe aai aerien Land earn be ee5eIy39 aiete1rpar e39t eade as aeaayienge mat nature tends lee equalize the kiaaetie aand petemial eneargiea 39threugheuat Itehe me tien In the above diseuesien we aaeeume Neewtena Law and deduced Haamailtam aa primeipale as a ED S q U E mE i The same aargumenat shears that Newateene law fe lewa frem I1amil7ten aa pLri1aeipe 50 these twee appareaehee In the elynamiesa 0W a particIe Ihe veetevriai and the V fiHtii 39n l 3fE equAiaalerat tee ene anether This reeaalet eemphasizes the essential ehaarae IBiI iEIiiiZ39i Elf v39aariataieaaaal prineipleea in Phj395iCquotSZ atheiy exmesa the paertienem physical 7 in tem1e ef E39 1 rEF alone Eweiatheut referenee to any ereerdineatae The Eil1quotl1mEM we have giiveen extends at enee be a system Di 1 partielea ef maassea man with peaaitieane aveetera l39Iaquot ltex1 ai I yarj 3Mk which are 39me viag under the in ueenee ef eeanaewative fercea W Here peaetential energy ef the syaaem is a funetiea Vxa1y1eaaM x aymez auerh mat EV 8V B E 1 39Eilte quotTi EquotET r y 39 SE 573 the kineiiee energya is X A E O D 9532 Z W E T392Zm dr Adar Ewe 0 11 and the amen twer a time imenral r P I 5 2 08 53 08 J T dr 39F1e In just the same way as aeeve we see that Neewmne eqjaaatiena ef mnataeieeen fer the aystaem I mi E F are a m eeea5aary eerImlfitE0n fer the aetien meA Thaavae a sataateienajry 39i I I i Hamilterra priermiple therefem39e helda far ETFW njite eyaeteeezm of prartiacelee in whieh the feareesa are eeansaeervaaetive It applies eqalaa y well to emerre EEEITEIEEH dynaamieaJI ayateame invemng eeaestraamte and n39ampgidf beadiiea and a se te cteenetinueua mezdiaie In additirm Hjami teaenquot e preineiplae can be made In yieald the basic aws ef Ea139E tTiCiquotl39 and maaaetisnme quantum taheaery and relaetia ityee Its in ueaee an premauend and fara r eeaaae1 1ineg that many SE i39En xE t regard it as the meat peeewerfm single prianeiple in mampathematieal physics and place it all the peirmae ie of ph39yaieaE aeieneee M a P 1anek the afeander ef quaaentum 39Ehe 39E39 expressed this View as I eHewa The highaest and meat emquoteIeed THE CALCULUS or VARIATIONS 529 aim of physical science is to condense all natural phenomena which have been observed and are still to be observed into one simple principle Amid the more or less general laws which mark the achievements of physical science during the course of the last centuries the principle of least action is perhaps that which as regards form and content may claim to come nearest to this ideal nal aim of theoretical research Example 1 If a particle of mass m is constrained to move on a given surface Gxyz 0 and if no force acts on it then it glides along a geodesic To establish this we begin by observing that since no force is present we have V 0 so the Lagrangian L T V reduces to T where T ml i 2 392 i 52l We now apply Hamilton s principle and require that the action rLdtrTdt be stationary subject to the side condition Gxyz 0 By Section 67 this is equivalent to requiring that the integral it AtGxyzl dr 391 be stationary with no side condition where At is an undetermined function of t Euler s equations for this unconstrained variational problem are dzx dzy dzz dt2 iG0 mdtzAGO matiG0 m When m and A are eliminated these equations become dzxdtz dzydtz d2zdt2 0 G G 39 m Now the total energy T V T of the particle is constant we prove this below so its speed is also constant and therefore 5 kt for some constant k if the arc length 5 is measured from a suitable point This enables us to write our equations in the form dzxdsz dzydsz dzzdsz G G G 39 These are precisely equations 6729 so the path of the particle is a geodesic on the surface as stated Lagrange s equations In classical mechanics Hamilton s principle can be viewed as the source of Lagrange s equations of motion which occupy a dominant position in this subject In order to trace the connection we THE CALCULUS OF VARIATIONS 531 the action ff L dt is stationary over any interval of time t s t 5 t2 so Euler s equations must be satis ed In this case these are H 39 I 9yrquot daL 8L 0 12 2 dt 8q 8q which are called Lagrange s equations They constitute a system of m second order differential equations whose solution yields the q as functions of t We shall draw only one general deduction from Lagrange s equa tions namely the law of conservation of energy The rst step in the reasoning is to note the following identity which holds for any function L of the variables I q Q2qm 142n mi dquot aL J m daL 31 ac L 3 d quot aq Eq dt aq aq 8t Since the Lagrangian L of our system satis es equations 2 and does not explicitly depend on t the right side of 3 vanishes and we have 3 LE 4 for some constant E We next observe that avaq 0 so acaq 8TSq As we have already remarked formula 1 shows that T is a homogeneous function of degree 2 in the 4 so by Euler s theorem on homogeneous functions With this result equation 4 becomes 2T L E or 2T T V E so TVE which states that during the motion the sum of the kinetic and potential energies is constant 393 Recall that a function fxy is homogeneous of degree n in x and y if fkxky kquotfxy If both sides of this are differentiated with respect to k and then k is set equal to l we obtain 9 3f yMmm ax ay which is Euler s theorem for this function The same result holds for a homogeneous function of more than two variables 532 DIAFFEREHTLAL ED39U L39TEUfHS In the fmllismwisnsg ssampls we iilustrate the way in whsi h Lsagr ssngE s Eqb1 IiDI39l5 can be used in svpsci rs dysnsmissfl spmblesnns EI l l39IlE Es gs psrticie of mass m mmrss in E mans under ths ins susnecs sf 3 g srss i tstin snaiE fsr39rcr nf smssgn i 11ds kmfr dirtf tes s t wsrd was nsrigsin than it is naturasi Es shsssss polar cwmrd39instss as the En Er 39rE iIE cmr39dinst ss 3 r and q M is sassy n sss that T msf392sf39239 4 93 am V km frquot sun has 39Lsgs1ng39isn is L T Av gsE Essa F snd Lsgrangs s squsstissnss are daL ass dr T U 5 d EL M quotE 39 15 uI39 58 J 3 Sinssss A L dsrsssr nm depEnd rs39p1isusiuy on E eqsuasmn 6 sfmws thssts SLME msrl is ssrrmsEsr m ss Jquot a T fur sums mnstanst sh ssssumed was be p ssitiwcs quotWe nsssst ubsstrvs sthst SJ can Eaisihi be swrj sns in me fEJquotT iiT r d H s 1 F 1 quot as s V pB is pirscisssligr sqgusltinns 21I2 w39hih we sulssds in Seuztis1 21 Isa sustain the scunsiusisrun that ths psth sf thus gfliii mr j a rwnis ssssi039ng Vsiristiannal prushlhems fsqr dimxhsles inrteggrslzsr Um gsnssrai msctshud of nd ing snssrssssry cnnditiscwnss sfmr srm instsgsai to bs stssinnary Lzsan be apssplied equally swell tin muiitipiss ixntsgrslsss Fm esszmgplc C idj if s rsgirn R in ms xyplam tmunded by as cmssd cums C Fiig 97 Let 2 zs he s El EIiDn P de ned in snd assumes prsssrsibed bsuundssy ssaluess an C but mhsrwise fbi39l39il39 a139 39 except fur the usual dis Eersntsisbiiity cmditinssJ Thiss funssisun can bs thought of as dss ning as vsrisb lss ssusfascse xed a mgs its bm1nar1w in sspsacs An isnstegral 0f the farm Icz H is H pk will haw values that d spsnd an the choice of 22 and we vssn puss ths prnlbilsms sf nding 21 ELmcti1m1 2 3 S39IH i L T39y39 funfticsn thss gives a sts39tisnar y value ta hi5 instsgrs THE CALCULUS or VARIATIONS 533 Us I FIGURE 97 Our reasoning follows a familiar pattern Assume that zx y is the desired stationary function and form the varied function Zx y zxy crnx y where nx y vanishes on C When 2 is substituted into the integral 8 we obtain a function 1a of the parameter cv and just as before the necessary condition I 0 0 yields W a W LHn m gJaoo a To simplify the task of eliminating 17 and 7 we now assume that the curve C has the property that each line in the xyplane parallel to an axis intersects C in at most two points Then regarding the double integral of the second term in parentheses in 9 as a repeated integral see Fig 97 we get 3 C1 X20 3 f immo lame R 82 C 110 821 14 This restriction is unnecessary and can be avoided if we are willing to use Green s theorem 534 n1mEmENrmL EQUATEQHS anal since p sf amp afI pX if P I 32 HI x AE Ti Szx H 31 23 it V1 pj 7 J I 53 32 A I becafuse av vanVis hes DH it f 397HGWquotS thay1 Jj X Harm cUntainlting 1 can be tmransf rmEdi by a similar p rDcedure and bcmmes We aw c nc7hL1de from the azrbitra ry natmte Of n that 111E brack t er d mpres5iun in 10 must vanish an f p j sf ids I 32 I F JJ 1 31 PG x dy is Eu1Er s EqL1aIin11 far an mrtremial in casse A5 b f f a statiunary ffunctintn if DRE existg fiszs an extreNmaEV that 5a39ti5 ars4s ms giquotLquotE ib 39U39 id I j c r1diIim14sV Example In its 5imple5t frzrrm H16 Kprv f m af Amimmaf smjfrlcesl was mrst pm p5e4df liw ELu39EEI as fn mws to 11d the surfaCe 2Af sn1afHe5t area buundedi by a gi39ren Ci39E39Edi aurm in spauze If we assume tharl mhi5 rvurvE 39p39r39IEJ i E1S duwn m a E105d MIWB 4 3urrUun4dinJg fa retgim R in tihe xy39p Ianei and al that the 5ur39f ue is ExpressiIampe in the fkmrm 2 2ryL than the PF blEm is to mimimiza th 5urfaEE area iniegral 1 in g EE ixquot 11 U subject to the haundarry cundit 0n mat zsry mwt assume hpre5 m39ib d LraIae5 inn p ELMer 5 equratiun H fm this integral is T87 zi 539 3 T BA ax p J 6 N P whieh can b w ri tter39n in 11hE tmrma Ii Emil 4 Eff El 13 THE CALCULUS or VARIATIONS 535 This partial differential equation was discovered by Lagrange Euler showed that every minimal surface not part of a plane must be saddle shaped and also that its mean curvature must be zero at every point The mathematical problem of proving that minimal surfaces exist ie that 12 has a solution satisfying suitable boundary conditions is extremely difficult A complete solution was attained only in 1930 and 1931 by the independent work of T Rado Hungarian 18951965 and J Douglas American 18971965 An experimental method of nding minimal surfaces was devised by the blind Belgian physicist J Plateau 18011883 who described it in his 1873 treatise on molecular forces in liquids The essence of the matter is that if a piece of wire is bent into a closed curve and dipped in a soap solution then the resulting soap lm spanning the wire will assume the shape of a minimal surface in order to minimize the potential energy due to surface tension Plateau performed many striking experi ments of this kind and since his time the problem of minimal surfaces has been known as Plrreau s problem Example 4 In Section 40 we obtained the onedimensional wave equation from Newton s second law of motion In this example we deduce it from Hamilton s principle with the aid of equation 11 Assume the following a string of constant linear mass density m is stretched with a tension T and fastened to the xaxis at the points x O and x If it is plucked and allowed to vibrate in the xyplane and its displacements yxt are relatively small so that the tension remains essentially constant and powers of the slope higher than the second can be neglected When the string is displaced an element of length dx is stretched to a length ds where 1 ds V1y2dx 1 y dx This approximation results from expanding V1 yi 1 yfquot2 in the binomial series 1 yi2 and discarding all powers of y higher than the second The work done on the element is Tds dx Ty39 dx so the potential energy of the whole string is 1 If VTf yidx 2 n The element has mass m dx and velocity y so its kinetic energy is myf dx 5 The mean curvature of a surface at a point is de ned as follows Consider the normal line to the surface at the point and a plane containing this normal line As this plane rotates about the line the curvature of the curve in which it intersects the surface varies and the mean curvature is one half the sum of its maximum and minimum values 16 The standard mathematical work on this subject is R Courant Dirichlet Principle Conformal Mapping and Minimal Surfaces InterscienceWiley New York 1950 THE CALCULUS OF VARIATIONS 537 noncommutative linear algebra in which division is possible The remainder of Hamilton s life was devoted to the detailed elaboration of the theory and applications of quaternions and to the production of massive indigestible treatises on the subject This work had little effect on physics and geometry and was supplanted by the more practical vector analysis of Willard Gibbs and the multilinear algebra of Grassmann and E Cartan The signi cant residue of Hamilton s labors on quaternions was the demonstrated existence of a consistent number system in which the commutative law of multiplication does not hold This liberated algebra from some of the preconceptions that had paralyzed it and encouraged other mathematicians of the late nineteenth and twentieth centuries to undertake broad investigations of linear algebras of all types Hamilton was also a bad poet and friend of Wordsworth and Coleridge with whom he corresponded voluminously on science literature and philosophy 17 Fortunately Hamilton never learned that Gauss had discovered quaternions in 1819 but kept his ideas to himself See Gauss Werke vol VIII pp 357362 CHAPTER 1 p p p p p p V 0 D t D Chm of the mazign fEE lfI i g themE5 Elf this bank has been thus iviea that only 3 few simple types at differential equaiiiinm can ha snlved expliwi ly in TEIIHS Elf lirmwn elEmEntarjy functions Snme f Ihcse typea are ds crib ea1 in the first three Jr1apt ers and Chapter 08 provides z dE taialed am39u nt uf gsemnvdf ordar Hnear wa quVatimns whom 3c1EuAtian5 are expressible in Karma of p wer EIrEglis i HV1Dwever many di wezrermal Eaqguati3nrs fall nut5ide these caVtegVGriS and nothing we have dune so far 5ugge5rtr5 32 prmEdurE that rnil391t w rk in 5uvh cases We begin by examim7ng the initial value pmblem dAesnribed in Eiemian JIW E f vixJ yrm W Vwhemk fxm is an arbAmaryV fU FICquottiDn de ned and c0 ntinu us in smne n eighborh0od of the point 1xmyjjI In e metmz Eanguxage vur purpmse to devise a memthod fair cUn5tructLing 3 funrztimi y yr wEm5 E graph passes msr ulggh tha paint xm3 and that 5atis e5 the di arential aq7ua tiiurn y fy in some gh b rh d Uf lg PE 98 We aw Vpmpagred fur ELIE THE EXISTENCE AND UNIQUENESS OF SOLUTIONS yll X y 2vI gugt 41 FIGURE 98 the idea that elementary procedures will not work and that in general some type of in nite process will be required The method we describe furnishes a line of attack for solving differential equations that is quite different from any the reader has encountered before The key to this method lies in replacing the initial value problem 1 by the equivalent integral equation yltxgt yl fltytdt 2 This is called an integral equation because the unknown function occurs under the integral sign To see that 1 and 2 are indeed equivalent suppose that yx is a solution of 1 Then yx is automatically continuous and the right side of y39X flx yX is a continuous function of x and when we integrate this from x to x and use yx0 yo the result is 2 As usual the dummy variable t is used in 2 to avoid confusion with the variable upper limit x on the integral Thus any solution of 1 is a continuous solution of 2 Conversely if yx is a continuous solution of 2 then yx y because the integral vanishes when x x and by differentiation of 2 we recover the differential equation y x fxyx These simple arguments show that 1 and 2 are equivalent in the sense that the solutions of 1 if any exist are precisely the continuous solutions of 2 In particular we automatically obtain a solution for 1 if we can construct a continuous solution for 2 540 DlFtFE1 tEi 4quotlquotitLEDUAT1EtN5 new turn Ultt e ti t tie the prebtetttt of etettrtittng g a pteeeea ef i It1t Ii That is we Tbegiztnt with a etttete atpvpr1etttimatieznt In a ealutien and imptwae it step by step by atpptlyiag a repeatabte tetperatttietnt twhiteh wet hope will ring as as eleee as we ptlease ta aa ettaet eetutiett Q primar3r advantage that 2 has eirer te is that the tintegral euqaattien pretride a a 39EZ tl fEE ti E39I1t meeztttanism feir teatrryint met this pt Dt39ES as we new see A rough appremmattien ate a aeliutinn is girvetn by the eenatarnwt funettietnt yt JI EVh whtiett E simply a heurieenyttat atrai liee threugh the paint xyet We insert this ap pteeimattiean in the A aide Df teqtuatitee pM in erdter tea tehttatin a eew atnd perhaps better EafpItptD iTl1a 1iiI31It yufx ae fetlewezt tr J The next ettep is an use yxi te genetate anether and KpEth lpa ewe better appt39eximattti en y11 39 in the saute wayt 7 tIn 4quot Jf 1U 9 9 the nth sttttage not the preeee we have quotI eta W5 J f39wnaf1 at In 7 pmeted utre is ealted PtECere a fite I xti T ef ttaeeeattitte EEJPFEJItH1 7tWIS39E We tsthew hew it werfka by tneanst et a tetw EK mPlE39Et The eimpfte inittital vafme pnrebiem 5 ya yt 1 has the elrwieua setuttient ytgt 0 e The tequivalettt integral equatien is 0 Mix 1 J yt tda Elimtte Pieard IE55 e1941 ene ef the meat erninent Freaeh rnathe1mtatttieiana et the past eEt I39ltt1 jp read tee et1tattanding enttttittuttette te anaI r395tLia ttte39tEted teat atteeeaaive Hp fttttt ti tati lit t wihieh enabled him te pertetet the tlteery ecf di ete ttal EltttlEttitIItS that Cauet1y had itt i39Ei3tEil in the l a and ma ffafntetta VIlhE l3It39fI7l1t yeatte Pieart1 5 Great Tih tftrt abeut the traaltlee aaettrntea by a eesrnptee ana3ttte39 mnettee near an easentiaI eingutatri tr which Lltae atimttltateId t39Lttquoth irtteertatnt teeeeteh ttewn he the present I tE51gr Like lrrue Frenchman he was at C t1tft39 tttSSIEtt r et fine teed tea was par39tie39ulatrt3r tend set b ettiltattiaeee THE EXISTENCE AND UNIQUENESS OF SOLUTIONS and 3 becomes ynx 1f0yn1tdt39 With y0x 1 it is easy to see that y1x1f0dt1x 2 y2x1J01tdt1x3C2 xp t2 V2 3 y3x1L1tidt1x f and in general 7 x2 x3 xquot yx1xE 3 In this case it is very clear that the successive approximations do in fact converge to the exact solution for these approximations are the partial sums of the power series expansion of e Let us now consider the problem y x y y0 1 4 This is a rst order linear equation and the solution satisfying the given initial condition is easily found to be yx 2e x 1 The equivalent integral equation is yltxgt1 t ytdt and 3 is yxgt 1 r y1ltrgt1dr With y0x 1 Picard s method yields 2 quot x y1x1f0t1dt1x5 x 2 3 n11m5Jw1xx5 0 2 3 3 3 x4 x 1 2 x 3 4 x 3 y3x1L 12tt2 dt 542 D1FaFeeeHTIiaiL eeuarmeess Me ill J 1 2 P LI P 3 4 E 1 g eel I3 1 i al xhx and in ghehnh erel xnE If 1 239r239r3 xW Ps P V I P fn 1 P P P This evidernhrtljgr CD h39 ETgESA Ie 1 3 25 x 1 ix 1 so egeehl we have the emet selutien In spite of these eibiamp ee the reader may nwt be en tjireh eeEwineed ef the pmetihcele ealuie ef Piheard se metehhehd What are we the de fr iheteneeJh if the eeeee539eive inteegreteieane ere very eeamplieeterL or net gpeseihle at all exeept ih pvrihhehiiple 2 ekeepteieism P fer the real pewer hf Pihea139dih he metehed hes meiniy in htheery efh di erentiiei equaehtiehmewnet in eetualily hnding eeluhtiens but in presving ugnder very gehnerel E39DI1i t39ii Ei that an inmel vahie perebhlem has 3 eeehxtien end then this seilretiezn is L1I1ieqLIEe Thheerheme that melee precise eeeerttiehs hf thhje kind are heelled E aw EFl39VE Ei end euneihquenese EhE erE39m 0hx shell ete1 e and peeve eevetrel ef these theerems in the next twee seetiens Find the exaeth 5eluteien efh the initial value pmhlem u u 0lv q Starting with y3x s 1 apphly Pi fl l methed3 ten eele39ule39I e rJe a3I1IJ fe ameli 39rD1 11paIE mhese re5uhe with the e1eet eehntien R1 FinEIJ the exam eelutiesn ef the ihitiha h v1 hme prehlehm 45 ZMLZTI5 IL J U9 0 L 1tza erti hg with vquot43I39i U eeeel eMate yyr y2e y3xj mxe end eemfpetre theee r eeu1te with 3 hexeet 5em iene 5 It is einetruzethieve he see how Piee rltrZE e methued weeks with e eheiee ef the rinitial epprxuxeiemeiiheh ether than the Feenetmle metien jmx39 39 m apply the methede In the ienitie vajuhe prelbijeim 4 with Ea W1 51 r 1 1 Ell me mu 543 THE EXISTENCE AND UNIOUENESS OF SOLUTIONS 69 PICARD S THEOREM As we pointed out at the end of the last section the principal value of Picard s method of successive approximations lies in the contribution it makes to the theory of differential equations This contribution is most clearly illustrated in the proof of the following basic theorem Theorem A Picard s theorem Let fxy and 8fBy be continuous functions of x and y on a closed rectangle R with sides parallel to the axes Fig 99 If x0y is any interior point of R then there exists a number h gt O with the property that the initial value problem y fx y yiro ya 1 has one and only one solution y yx on the interval x x lt h Proof The argument is fairly long and intricate and is best absorbed in easy stages First we know that every solution of 1 is also a continuous solution of the integral equation uuufnwmwa m and conversely This enables us to conclude that 1 has a unique solution on an interval Ix x lt h if and only if 2 has a unique continuous solution on the same interval In Section 68 we presented some evidence xv vyr RI H gt 0 4 4r all FIGURE 99 tJtFFEREHTt L E U39U33t39Til UNE 5ugg1esting that the seqttenee elf funetiertte yx de ned by ytttx E Jrquot39t nt 0 Fe p 6 Ft T 3 tt 45 2 ms ere J fits Jett tit veentvergtes tn 3 eetutien of 2 We nettt ebeettte that ytc ie the nth pertitett sum ef the seirtee ef f39Ltl39tCi39DItS J39FISlI Jquote397 E J l t39IN 1 J tt39s PitI39J 1 4 text tu x3ttl tam t vMtI 4 G 4 zse the eentretrgence et the EEIQIIEHEE 339 is equ39i39veIent ttu the e39erwergenee emf thit eettee In tm39Ie r te cempIerte the pteef we pvtedttuee at number tt 0 U that tle rtes the itntttervat L E 514 S tr anti then we thew tthet en this tItIEt r39at the ftetlte wing stetetttentss ate tttrtte i the 5BI t E53 4 eirr1nverge5 he a fttnetLiee y39t39x339 it 39 t is e E lnI mtmD VuE eeluttern et 2 iii ytxf K the e nl1 r ee n39titnuees sttiutien at 2 The h quotpUT htE 5 5 of the titteerem ere used te predttte the pestittee nutrtttber it as feltewe We have eseurirted that feyj ettd eifttay are ex etieeetue fMquot EE UIm1 D E thee r39EEm HWIIETIIEI But R is eleeed tfie the eesnee p it itquottti ttdEf39S its heutttIert ettdl tee1tn ded se eech ef these ii 39L1Itl1Ti3939iCtft39t5t is neteteermw bounded en means that there ettiet eene tat139te and K such that lfftwttltt 5 M 5 and L 5tt tm ell peters eyt in R We next Ctb EI tr E that 0 ltt fl end xiy2 ate ttietirnet D t t in R with the El tt ex e eeetdinete then the meett 391t39 JLtE theetem gueterttees that V P tttwtt amp team 2 arltxtt 3 18 Wet tquot fest same mtmltet39 ye b etereen y and It ie eleat ttem ti attd T ttttett tflilw39 E fixwft 5 Me M 8 if r tttttt pein39t3 iftt3j393 ettdl 3J in R tdistittt39t Dr met G lie en the same wetttieet titte We mew v thtttt it te he em pt tSiIt tquotE ttumbet such that E 4 1 9 THE EXISTENCE AND UNIQUENESS or SOLUTIONS S45 and the rectangle R de ned by the inequalities x xul S h and y yol S Mh is contained in R Since x0y is an interior point of R there is no difficulty in seeing that such an 1 exists The reasons for these apparently bizarre requirements will of course emerge as the proof continues From this point on we con ne our attention to the interval x xol S h In order to prove i it suf ces to show that the series i 0xi 1x quot 0xi l 2x quot txi quot yx y ux 6 10 converges and to accomplish this we estimate the terms yx yx It is first necessary to observe that each of the functions yx has a graph that lies in R39 and hence in R This is obvious for y0x 2 ya so the points ty0t are in R 5 yields f ty0t 5 M and lYrx quot yol f u drl s Mk which proves the statement for yx It follows in turn from this inequality that the points ty1t are in R so ftyt lt M and iy2x yoi ftyt dt 5 Mh Similarly xflty2t ml 2 Mk 10 i 3x Yul and so on Now for the estimates mentioned above Since a continuous function on a closed interval has a maximum and yx is continuous we can de ne a constant a by a max yx yo and write yx yoX S a Next the points tyt and ty0t lie in R so 8 yields If tyt S f ltyot S K y t yot 5 Ka and we have lyzlx min fltflryltrgt1 fryoltzgt1gt dr 0 S Kah aKh Similarly ifit 2t quot ft Iti lt K i 2 quot 1 l 5 Kzah S0 i 3x quot 2xl 3 fr fty2t my1ltrgt1gt an 5 K ahh aKh2 G EIFIER ENTIAL EfJ UAT39DH5 W rmnwtinuing in tfhiis manner we ml t ha t nmrra ymf l quot5 t 5quot for every n 391 392 g Each term n f th 5er ie5 IELUA 15 therefore E555 than 0139 equal tn the c r2resp ndinVg term nf the sariE5 cjrf Eit3 I39IJEIaIItS W at aKh e a Kht N But quotI guarantEe5 thaf H15 SE iB5 cnznveltrge5 an M cUnvergE5 quotby the 0n1apari5 n E5I 4 cunvergas m 3 Sum which we dei1mm by y39 and yx yxj Sin2 the graph of each yfx lies in R1 it C evident that the gruah uf Mr aim has this prn per39ty Nnw fur the pmof of ii The a39m0 vE arumem shnws not mtly that y ir wznnimrges ta r in the im tgrvaL but align th ul this mnvergEnve is m f0rm This ITLEEHIS mam by ch Us39ing kn ta lbs su iuziemly large we can main y x as alas2 as we phias e In 39I far an 1 in Ihe VnIErumT Dr l I1D1 preVciseir if E v G g iven than thyeire cf3i5I5 a p Si39rEi inte gar P14 5IJ1chA that n nu we have yVxJ Vyrr lt2 E Arm all if in the i n ter vaL Sinus eacih y xV is nTiearly cmzz nuuus Lhis uni fm39mjiVty 01 the CiD n1H l7 EnCE imgplias that the function x is salsa c mtinu u5f To prove that yc is arcIuaIljy H smrutinn of 2 we quotmust 5hnw that NH hffUNWmamp an But we knnw that yrm fIyatr11 d 1214 510 5ubtrac1 ing the leftquot side DE 32 from the left side Elf 11 isms S Mn EfiIJW df J 39 M Mt ff LFIW fIIr AfW394 t and we mhIain pV l Mn fmmmw ii 5MU JUVJjUEhUH fhHHD e We will nut discuss this in dEIBiL bvurt the araasnamng is quite gimp and regtg can the i3nevquamp4iitjy Iyli W IAyx EV mil iimr mill fjmfif T MEJH E i I quotH 39quotTn quot39EN I 1 EJMLII 39 Jmff l in E39 VFf ilVA THE EXISTENCE AND UNIQUENESS OF SOLUTIONS Since the graph of yx lies in R and hence in R 8 yields nn nwmM4 lt yI yxl K1 max y x yx 13 The uniformity of the convergence of yx to yx now implies that the right side of 13 can be made as small as we please by taking n large enough The left side of 13 must therefore equal zero and the proof of 11 is complete In order to prove iii we assume that 7x is also a continuous solution of 2 on the interval Ix x lt h and we show that y7x yx for every x in the interval For the argument we give it is necessary to know that the graph of 7x lies in R and hence in R so our first step is to establish this fact Let us suppose that the graph of 7x leaves R Fig 100 Then the properties of this function continuity and the fact that 7x0 yo imply that there exists an x such that lx x lt h 7x yol Mh and 7x yo lt Mh if Ix x0 lt x xl It follows that D7051 39 Yul Mh Mh gt M ix quot X0 ix 1039 h However by the mean value theorem there exists a number x between x and x such that U ly xl lflxgt7 rl 2 M since the point x7x lies in R This contradiction shows that no point with the properties of x can exist so the graph of 7x lies in R To complete the proof of iii we use the fact that 7x and yx are both 4l xR yoii quot quotquot Hll FIGURE 100 imFFEREN139AL EQU sTLlUHS selutiens ef 2 te eeieme Wile W39fJil J msee rmrsyse as M Since the graphs ef x and yr heth he in 0 LB yieds eea H7zn he Eh mes i sla E vrJl This ifhIies that mes i r E ywrVi39 fer emererise we wse eld have 5 Kh in E W1tFElljiEIi tea 9 1 rflehewsh that IIf w39 ss yx1 fer eeerfs I in the interval I 5 it end Piesri s sheseertesm is fully preved SD Remer39k 0 This theerem eeh he SIlquot El lg39 7ilE iEd in various ways he weekehih its hypethes es Fer siesIenee eer essui39r39hptien that fay is eeminteeus en 9 is stsehger when the preef requires and is used e 1yquot lie ehteih the ieeuhshty 84 7 eeh therefre ihtr e d7eee this mequslity ihte the theerem es en essemptien that repisees the eee sheen 5 39Sy In this way we esrhivse at s sttreher fesm ef the 1heere1fn sisnee there ere many fenemiehs that leek e eehtihueus perIia IjETiiV3tiV tE ELIE hseeeertheIess setisfy39 3 Lfer seme eensIsm This ineqeslhhy whielth that the ei erenee quseetiese rye fs rJeJ Jquot1 5 J is heended em W is celled 3 L psehz39 squot eehdiriee in the serie hIe ye R39 ErI1 l k If we dI ep the Lipsehietz eendiIieh she eshsseree eel theaquot2 fey is ren1iesuees eh 5 then it is still peslsihk te jpreree that the ihitia1 value preblem hes a seIutiee This il ESllIE J knewh es Peehe s n eer ee1 s The only ikhewh p1reefs depend en mere seepshisIiehstsedF srg u 3 RrudelE LipsehiEe h1EZ 2 19 DT was s prefesserr st Been Fer mess ef his l ife 1 lie is rememhered e hie 3 fer his reies in simpEifysieg she eflstrsifggi11g Cssuehquots eriigjhei 1 heery ef the esisIenee efhe uhiquehwess e1fseIetiens ef eif erehsiell eeue tiens Heeeeer he she esteheee irie1h eIquots sheerem eh the rsepseseet39sh iliftfy etf e Eem1iien hy its Feerieers sesfiess e39hr sine ds the Ferrhe e fer the mlmbes 1W3339E pnesmeer hstegerr eeh he eepare sseequot es s sees ef feer seueres es s eehseeeenasee ef his eeei Iheery ef the Es teri satiesn ef inEegrel qes39tersniees she made usseefel eeetrihutizehs rte IheeretieeI meehehies the esletees ef ssrie tiehs EesseI funetie ns squed3rstie dki eresntie39i fI39Z1quotIl39I39Iquot39lIE end the sheerye ef ese5 ehh 4 f3 eiseeppe Fsane i f ifiW32 mien 39Iegieien she e1s39the39metie39he streJeglf39 tin eeheecili lIhhert s eeiestrrseee IIrlteehree39ns ef elehe gesememr shad the week ef Whtteheed ar Id Resse M eh reeIher39eetiieel Ilgse His ees teetesh Fee the hesriIisee integers see es gehereeees ef stesIieh ts te wxender whether39 ell ef medem elgehhre z serhe Isinal el39 eehsphee Ate sener the zeheiees sehseuee i39I39 is huestlj In 1391 he a T Ll ded the me7t39hems39lhs eeehl wish his remerkehJe eees39tseetien ef s eentiineeess eeree in the p Isees thst eempiletely lls the squsr39e J 3 5 1 H 3 p5 1 hUh ferI39uee1 ehe39 fer e mien efhe eshmeel legie se hfihghhw his l e preef ef the ehnveee esrislenese theerem Eierr seileutiehsv ef 1trquot fIyJ wehs inedeeuel39e end e setisfeeserg r yprueef eves nest I euse until many guests lesser THE EXISTENCE AND UNIOUENESS OF SOLUTIONS 549 ments than those we have used above5 Furthermore the solution whose existence this theorem guarantees is not necessarily unique As an example consider the problem y 3y 0 0 14 and let R be the rectangle Ix S 1 y S 1 Here fxy 3y2 3 is plainly continuous on R Also yx x3 and yzx 0 are two different solutions valid for all x so 14 certainly has a solution that is not unique The explanation for this nonuniqueness lies in the fact that f x y does not satisfy a Lipschitz condition on the rectangle R since the difference quotient f0yf00 fquotf 3 y0 y y is unbounded in every neighborhood of the origin Remark 3 Theorem A is called a local existence and uniqueness theorem because it guarantees the existence of a unique solution only on some interval Ix xol S h where h may be very small There are several important cases in which this restriction can be removed Let us consider for example the rst order linear equation Y 439 PXy Q0 where Px and Qx are de ned and continuous on an interval a S x lt b Here we have fxy PXy Qx and if K max PxI for a S x S b it is clear that lfxyi fxy2 IPxy1 2l S K lyi y2l The function fxy is therefore continuous and satis es a Lipschitz condition on the in nite vertical strip de ned by a Sx S b and 00 lt y lt 00 Under these circumstances the initial value problem Y Px Q0 x0 Y0 has a unique solution on the entire interval a S x S b Furthermore the point xmyo can be any point of the strip interior or not This statement is a special case of the next theorem 5See for example A N Kolmogorov and S V Fomin Elements of the Theory of Functions and Functional Analysis vol 1 p 56 Graylock Baltimore 1957 DiEFFERENTTh LL ED UVA39TID NS Thmr em P Lea f ry be as cnI39fn uom Afunmom lhm mrn395 e5 Lip5rhi z mm n39rri quot Pg 5 J I I quot quot39i2I raw a 539ir p de ned by H E 1 E b and m r i ff r yE 3 n5 mEn39a f IhE wrfpt then the ErIafiu ua39x39u probfern my y E Y km EM M has om and anxfy H 1iquot 3 I39H39If y x an 139I re nrErvm39 5 x E E1 EraHf The ar gumenI is similar in t hat Agivaen f fir Theorem A1 quotwith Certain 5impi ca39tiuVn5 p rmitmd by the fact that the regi ne under di5Eu55iinn is Ann bvzgrundedi ahawe DIE b l wa In partfi ular we start the pmuzrf in the same way sand Eih w prthat the sE1quot39iE51 4r and therefnm the sequence u nifcrrmly cnnwErVgEnI mm the whulle iflTl39I39E7 l 71aquota1 H 5 xi 5 Err We azcmmpilish this by 39u5irrIg a Emmewhat dai rent mcthud1 of esti mMing tlm terms of 1ESETEEES ID First we de ant M M arm g h M my g Vmax iliyxJL M M and we n0tiE 6g yHVx 2 M and LMIJI ynflxll E q Neat ifIn u J 5 b it fmIl zrw5 that iy2I AMI E J farm t aw Ii 139 0 b rlfrnW f1rALJMdr 539 V quot 0 it E KM39n39 31 M g A wyzim rEm1 a C I Im a mm as 7 A e I stall 2 3ML 39 x dai ant rim gEnEraI I iiInVIlquot li The 5am e argum1 Ent is awn valid far at E 1 E pm39viAdEd only that 2 19 is replfmad by ix E x an we have llI E Jiulin I Lyrrlix yr U F1 i S lib 339 39 39 r l H A K M H 1 THE EXISTENCE AND UNIQUENESS OF SOLUTIONS 0 for every x in the interval and n 1 2 We conclude that each term of the series 10 is less than or equal to the Corresponding term of the convergent series of constants b 2 3 MMKMb aK2M 3391 K3M 3i so 3 converges uniformly on the interval a 5 x 5 b to a limit function yx Just as before the uniformity of the convergence implies that yx is a solution of 15 on the whole interval and all that remains is to show that it is the only such solution We assume that y7x is also a solution of 15 on the interval Our strategy is to show that yx gt 7x for each x as n gt 00 and since we also have yx gtyx it will follow that 7x yx We begin by observing that 7x is continuous and satis es the equa on rec yo flfty39rl dz IfA maerl7x y then for x S x S b we see that Iy x yxl X flty tl ftryt an s X lftYtl ftyofll at Xu s K mt yiidr lt G X lquotI zXl ft firm ftxylt dr 2 K I Mr ml at KZAJIU xdI K2A X0 and in general x X0quot ly x yxI 2 W H A similar result holds for a S x S xo so for any x in the interval we have rt b n mac ma 2 mi 0r lt m n n Since the right side of this approaches zero as n gt 00 we conclude that 7x yx for every x in the interval and the proof is complete mFFEuEmtuL E UAT39I NE K Let rm be an atbitrarjt paint in tile plane and D39I15id I the ir2itial value ptublamt y 0 J gr ytxm ya Eitptaint why Ttheureem D guarantecsg that this prubletn has 3 ui t39iquB 5Gilu1tiD Dquotn Snme intewa39t Ix xa E Since fIIiy y3 amt SfIquot3y B 31 C TlquottIfiI39il17LDl15 DH the entire plan e it is letmpttinug tux mntstude that this sulutiuitt is ua1id fmr all 1 By ED SidEI it391g the 5uu tiuns thmuh the pn int 5 MII39Il39jIt and l39Tt shnew that this cmtncf usi n 5 sumettuim s true and sumEtim s faal5 1i3I 1d that tthegr furet the lElTfIf39I39h quot tquotIaquot39 tE is nut Ecgitximatet 0 Slhsorw that 4 Kym ailA dues not sattistfy a Lipschitz tutn dittitunt am the rtactanglua Ix E 1 and Ill 5 y 5 1 M dm5 satisfy a Lipsw39t1itz I dIi39I n urn the rectangle Ix 5 1 anti c E 6 E d where U 2 p 3 39 hI lw that fry 1 M satis es ai Lip5 chitz ctutrtdititm P C mctaugLlE J1aE I and y E 391 but thayt fails tu Etf5t at tuangr puiuts ut this IquotEE 39Ig1E5 P Stht w t hat xytl Era 5ati539 e5 a LLi p5dE39hitZ CDt1 itJlliEi DI1 on any rectange at S r E b and c 5 y E d h darts nut 5a139mi EfY a LtiJpschiIt2 t 39 dili Dn an army strip E E 1 5 b and m y m Shaw that fty xy 3 5atisr e5 l Lipsctlitz Egandituitun n f rectangie a J E E1 and C 5 y E Eb 5atis es 3 Lripusrchitz cun diutiun un any 5t1quotip at p I 0 b ant tr W 2 cjlt dsores mm a Lipschitz cut itziunt un tha EH39VEri 139E plant 6 Cuntsidegr the initiatl value tprublcm 39 IN Jw xut Iquott aft Fm what pnin t5 xtyn dues Theurem A implyquot that this prublcm has 3 unique SCllJIiEt Du SDWIE ii1Tl EEt1WE1 Lt t39ut E M M far Whatquot paints tre 194 tfluast this gprubtem avutuatlIty fume a unique S luti rfl DEITTI Sume i n tEW 3i 1 ml IE AF 39 39539 Fur what yptrints x y dtuE5 Theorem 0 itttpiIr that the jlfl ti it quottr UE prublem I J U r yxtn Im has a funtiqt1 e sultutimia m1 sumet itntti er valt H jtigi E u 6 pW 0Y Ptittatdu method at 5u c ce55Twe appruumtatiutn5u Can 3150 be appliezdt tran uysutems uf trstt order equattitutnut Let us E f 39Si IEquotr for exttamplew the tinittal value ptuutem cunsisttint gt of the ufulluwtng pair f rst Dtd t Eql1 IiD139IiS and THE EXISTENCE AND UNIQUENESS or SOLUTIONS 553 initial conditions d Ty fx Z X0 ylb x 1 dz 3 gxyz zxo 20 x where the right sides are continuous functions in some region of xyz space that contains the point xyz We use the differential notation here in order to emphasize that x is the independent variable A solution of such a system is of course a pair of functions y yx and z zx which together satisfy the conditions imposed by 1 on some interval containing the point xo As in the case of a single rst order equation it is apparent that the system 1 is equivalent to the system of integral equa ons yx Yo lxfltytzt cu G5 lt2 zx 20 I gtytzt dt in the sense that the solutions of 1 if any exist are precisely the continuous solutions of 2 If we attempt to solve 2 by successive approximations beginning with the constant functions ox yo and 301 30 then the Picard method proceeds exactly as before At the rst stage we have yltxgt Y0 ffl Y0 301 dr zxgt 20 fgryoltrgtzoltzgt1dz X0 at the second stage we have y2ltxgt Y0 Xft y1t2Zltl at In z2ltx Z0 X gry1ltrgtzltrgt1dr and so on This procedure generates two sequences of functions y x and 554 mFFEHEraT 1a EmLJmms z a and under 5a139itabl hypmheses the arguments f Them1rem 0 F can easily be ad zmpted m prm that these sequ nces canverge in a 5lmi0nA Df 1 Aw39hic h egxists and un ique on some interval Ix xul we now spmialize In a llinearA 5ystem1V in quotwhich ma funCti0ns fxyVzI and g xyz in 1 are lineaar func39t4i0ns emf y and 3 That is we EDEEMEII an initial vialuc problem mf the fm39m If E pr xly 81 mtg Ph 5 pgmy p 2 O where the six functi ns px qr aand rj px arc cm1tinun us an anw in1ar39vaJV m 5 3 E b anvil AI is a paint in this interval SiA 6FC E E h Uf these fur1mim ns is bmnnded fr u 1 1 5 there EKi5 l S sag Arzunstant such that jpx E K and Iq39xl1 E u far 0 1 d It is fICiW easy Vtso 536 that the functions on the right sklres 0f the diEerent iai Eq39Ll ti E E in 3 salzisfyi mndi1i0ns of the fm iIquot l 39 fIy2s3Vl 5 KU 1 E P25 39539 I31 E 32 and 5 Fzl 393 Just as in tlw pmuf nf Themrem o these mnditi ns cram bE used to Shaw that 3 has a Ll iqllE 5nlutiz0n on the Wh 3 in39tewa4l a E 1 E g Agiain we apart the r EadEr th detaij 5 Thes4e ArEma lts abuut systcms make possible ta give a sismple proauf Elf the m awing baesirz tham am whirch we stated at ht be ginniAng Elf C hapter and whiclh ha5 Tpw ayd an umbtrjusiws but cxrL1ciaI rule all nf muquot w rk DE sarmnd FdEE linear equat iUnVsi Theorem LEI PJ tQxfJ mm R t he zrmztinunm fimcmns an an imem f 1 5 2 E If 1 is aw pain in rmquot fM f FU r d quotn EM 3 Hr H39 Vnur39n JEr5 wuH fE 1J Er39 than Aihe iHEIf l39 Mame p39r b39E39rI viii y M az Mz y PEIJE QI Ov HLIL ym quotn and LP M34 yr 4 hr13 ww and mn 39y may 3mfu 7mn y yJV rs rm nr rum39 1 5u 1 5 b Pmaf If WE intmd39u ce thug quotIv Ti 5ME 3 A then it p Eltar that mveryquot 5CJlJ39L1Vfi 39Il1 Bf 4 391p iEd39Ei H 511 In39ti39n atquot the Hnear Vsyst tn1 2 3 39m was VPC1332 quot QII F RU 2I 1 Ha arm EUE1 HEISEE We have See39n that Q5 I1as1 wumqlle E hJI i39 an W13 mterml H 2 1 50 the same true atquot H THE EXISTENCE AND umoueusss or SOLUTIONS 555 PROBLEM I Solve the following initial value problem by Picard s method and compare the result with the exact solution dy dx z y0 1 dz 5 y 20 0 CHAPER 1 f BY P b S N D jpmArmmr s Mur1 mmg fcsf SE39 39Er1 E5 P u MHfrary Academy Wes Pmfn New Ycirk Ir99ri391335 T11 pZ Despit HIE bmad range mf p mVwerful anaEytical tioms pmsentad thrm1gAh mwuvt thuzi s hank mar139 GECasinns cry Um E0 the appI icam1n f nVum ericaI memnds f f s01ving rdVinagry di V eren39tiafI EquatViDns For exame5 an axanct snlutiDn may ha unav39ailafbl e Dr may be f li ttla practical value PN situati39m n Dcwrs Wham power stries LEDl39H i S In ina mr sEmnd GrdeI EsquVati uns am EquotU S tI39UCtrEwd In g naral tha Series are r a1he r g fnuzl aVppr ximatim 1539 near the initial mndgitinn but the T3l0r EKpansiUn3 can sum TEq1Ti139E prUhibiti vely mAVan39 terms sh uid the su1l4u tim1 lime required at same large distance fmm thagt point FDir large systems uf equati ns an Exact sUl uAtiur1 n may exist in VECEDF farm but the 5ub5 aquEnt algeibraic mnanipulati ns may be uvejrwhelm4ing F urtherm re5 nuVmeIical sojlutinns Fmr 3 dVtailed h1fi5t nrial aIci1unL cf the imp rtana min plaa39ed by rm apKpA icaEimn DE n39umeNTcal mehuti5 n dEfferurmEial qg1maIi JrJ5 25gr Garmtt BMhn 395 quotNun1trLiaH F1uicl l J39rnAamicsquot hE39 1931 JQhnwn Numa1n I ctuw putEisheuif in S MM R7E39UI39EW ml 134 uI939E3fa Ei NUMERICAL METHODS 557 should not be cast in a light of last resort for they form the mathematician s petri dish a crucible in which he can conduct any number of experiments on his differential equation and by proxy the very thing he is trying to model These numerical methods rely on two fundamental but distinct approximations First a differential equation is replaced with a difference equation and the role played by a continuous independent variable is then assumed by a discrete one For this approach to be of any use it is important to understand the conditions under which the solution to the difference equation is close to that is converges to the solution to the differential equation Second in virtually all digital computers in use today the realnumber line is approximated by a large but nite subset of rational numbers Limiting oneself to only a nite range of rationals can have unobvious but crucial consequences in certain cases the errors made by the machine may indeed be catastrophic At any rate both of these approximations permit the difference equations to be implemented on an enormous variety of computing hardware Nevertheless there are many apocryphal stories told of engineers performing expensive com putations on big computers only to obtain nonsense answers We emphasize here that existence and uniqueness questions discussed elsewhere in this book are vitally importantand should always be considered first Beyond these other problems such as numerical instability and the existence of spurious solutions can cause dif culties Despite the abundance of welltuned algorithms for solving ordinary differential equations the reader should carefully remark the need to be evervigilant Before appealing to the machine for aid it is always wise to know something about the answer one seeks That is the practicing scientist should endeavor to know as much about the solution as is possible For example is it bounded Stable Periodic About how big or small should the answer be Careful attention to these issues as discussed in the preceding chapters will stand the reader in good stead for what follows3 zln 1965 N J Zabusky and M D Kruskal discovered solitons in just this way By considering a particular version of an equation governing the motion of surface water waves and experimenting with its numerical solution they deduced the existence of mathematical objects with truly surprising properties Solitons and the differential equations that govern their behavior have been one of the most intensely studied areas of applied mathematics during the last two decades 3 For an excellent historical background on the evolution of numerical methods for differential equations that occurred in the decades surrounding the development of the rst digital computers see Herman H Goldstine The Computer from Pascal to von Neumann Princeton University Press Princeton 1972 p IIFiFEREHTtAL EEJ39UATitZJt4 E In etdietr te usdtemend what we InEt ttn he s numertiest setutiean ms 3 dii eretntia1 eustient we eensid er the simple inittietesstue problems Q 0D he v L 1 PN premem hes the ehviees seletien V e tend fer misses t hE retiQal pursptstes this is eneugh Htiwevtersi in s prsetieal epptiestiene it might bg I IECES S3 l3939 tea knew values eff the sehttien when 9 057 and 113 deeinlei 1649 is ihely tel he mere useftti than tthe s itntle0 em In E trag its the theeret1ieel setutsietnt set M s tmmterieteli S ifLlI39iDt391 see he ptesided by a stable ef sstt39uest fer e er s pe eLtet esteutstert Either way the number es iehteined elepetmzled en sees tttiewultedge st the terrnutls y a is In this ehegjster we GitBStCI ii3E several mettheds of esteuisting an ei4pptetsittmistiien nurtterieei seiutiiert trrf the ferrn Jr 2 I stern he sum We shall essutfne that this ptquotDb iEm has is untiquie tsetutien dtenested by iyx 39Oer methods CUt tSiiEtI sf e eemputetienetl pr ted fES based seftety en the fi f fmamp39iDE given by 2 and are eetmpitetel1gt izndependeint ef whethet s f39tTtT lL1ei fer MI is known er insets These WLtmEr iiE i merthedsi arid others like them ste thereteirte erstrltemety eetushtte fer these iniitistl eettuet ptreha lems thsteat1eet he selved eseetly end else fer these ihssitng exmet fiermst seiurtsiens that see jprsretties tly iI1tf39ampEi b39iE4 Let 0N he is titsttte mete isspeci es about the natures eif these methedTsi We shall not appresimstte the eseet sealutiien yitt fee all setlttesi set 1 in seme intewslt hut Uni for a discrete seqsutenee ef pecints begittnirfig at I g say En xi xi xg v XII h Ii t r s1 4 Ft where h is a positive number This means that we want en epptestimsti nt yi te the esheet melee yfxstljr en sppreszitimatiteta G to the exact s39alue eyts2J and so es Eseh nrumetieaiI metthed we deseti b e Witt he he rule fer esintg yamp te es mptttte yk5 Sinsee we knew the intittisl values ys ya this is esset we we apply the rule witht 1 te ehtsiin Ayth with n A 1 its eltain ya ere Wut genectst purpese is te sipphr etmtlgh estquot the detsils est each imiethed the enshie the reader to sepsis it fer himstetf39 if the tteiett sheets tester isrise tested tIE39il3iiS testing with the piethrs et esemputttiinig rt1sehitntes and pregremtmitng ltsngusgeist fJDI several teasers quotThe neted stmeritesn msthemstieiee G W iiemmin seitl thsti the ptlrp st ee39mpuIir1g is inisights ne t vnutr1391bers Eses see it tshes 1r T1Diquot39t 39 ti1sn insigTht te hui3939J1 e Si39Ejquot39Etl I39epE t39 er s spsee 5i39tt1t tiE 5 These are s Cst ed s i39ttgisstep mesthtedtsi There ere else esriees rt1u39ltistep tnethedis in whith sh W deipenid s net test on ye tmt pessihlriet1 j 5 snsi esrlier termsi NUMERICAL METHODS 559 First those issues are best left to specialized texts in numerical analysis Second it is our experience that virtually all students have some familiarity with computing fundamentals and should be able to write programs where appropriate to perform the calculations required by the exercises in this chapter As to the means that is better left to the student and his teacher Third advances in computing continue at a dizzying pace and we see no need to burden this book with nonmathe matical details that might well be obsolete in only a few short yerars We shall illustrate our methods by applying them to the simple problem y x Y y01 3 which we call our benchmark problem This differential equation in 3 is clearlylinear and the exact solution is easily found to be y2equot x1 4 We have chosen 3 as our benchmark problem for two reasons First it is so simple that a numerical method can be applied to it by hand without obscuring the main steps by a morass of computations Second the exact solution 4 can easily be evaluated for various x s with the aid of a pocket calculator so we have a means of judging the accuracy of the approximate solutions produced by our numerical methods PROBLEM I Have you encountered any examples in other courses where either the textbook or the instructor referred to numerical solutions of ordinary differential equations Give an example and discuss what you read or heard 72 THE METHOD OF EULER If we integrate the differential equation in 2 from x0 to x x h and use the initial condition yx0 ya we obtain imnyuaf7naywu Of I yuoxfeww 3 Since the unknown function y yx occurs under the integral sign in 5 we can go no further without some sort of approximation to this integral Different types of approximations correspond to various meth ods for numerically solving2 560 ttFtEm ttuterttL EID UtTl Utt Ettttlettt methmt is mbtainved fmmt Ih simtpESE way of ap pt timtatttting the inttegrtat in 5 It is wurtth mnsiderring because it ptwes that way for ant utndeirs tanding mt U HquottE morn accurate tbtut more c mptictated tttEthUd5i Th idea i tn obtatitt y1 sszrurt appttnxtimtati n ttte ytt b y assiumitntg tthatt th inmgratnd xAty in 5 vtarties 50 ltitttle Gquott tEfr the intterval t39 1 5 x thart uttly 3 simaM rt r is rnade by rep Iacing by its vmu f xwy 4 at tht left ettdpD1inft This is tquithalentt to reptacting the ii EgI39 d in 5 with its EEmILh t1rtttte r 39T tayInr ptulynomital that its RC1 3 J t J39ED L M EJquot39 I Ink wthere Z is X Taytmtr t39E39rttaitdEI39 Ii EI m 3ffjy and 1 c 5 r N t illtlg that N f y we sttbstiltute 435 into S ta l3tain WhEFE a ya 3 Ft 3939 ffxtt tJquott13quot vm We EdIgtlPPUquot that u is srnttll in an appr priat E sense and EglEltC I the Ph Haw srnalil is Small in gEiFIEI tEIL amiE mar p fIiEUl if39Is39 Wh gd term is small are irnp t tant issuttes that wtittf be dt5cus5e in mtnt39e dctarit tater SE Fm bIEnt 0 c SLEC Ii I1 3 quotfor 3 related tdttcussitrrng Negt ssting this term we htave t tf quottI Y 1iiIu f39 quott1tv 7 We now wnttitnjuet and b tain tfmm y in thte same Away by the f rrmtula ya Vytl xt1y and in eneratt we ham ma yt hfxwtt 4 for P 1 Y nci The gmmetttr ic n1eitttittg ml thEse ft1rmutas is 5hcwn tn Q t tt wtmre the Emat1t h curve T the lLJ 1 t1DW nt etmc39tt suiutimt which A WQ Et t t H39T at tatcm1 t Isqn tEmtr 15 rst step t HE 1M NUMERICAL METHODS 561 is being approximated by the piecewiselinear curve generated con structed from 8 To understand this gure remember that f x0y is the slope of the tangent line to the curve at the initial point x0y The point y is found by constructing a line segment beginning at x0y0 with that slope and marching it in the positive x direction a distance of 1 That point becomes the second approximation to the solution The gure indicates the vertical distance between the solution and the approxima tion as the error at the rst stage An important quantity derived from this is the total relative error 1573 at the nth step de ned to be lyxy Equot yx 5 This quantity is often expressed as a percentage providing a comfortable way to gauge how accurately the numerical solution is performing Now using x1y the process is repeated again to obtain the next point at x2y2 also shown in the gure The geometric realization of the Euler method suggests that error can build up rather quickly which is in general true We illustrate the Euler method by applying it to the benchmark problem 3 We approximate the solution at the points x 02 04 06 08 and 10 by using intervals of length I 02 It is convenient to arrange the calculations as shown in Table 1 In the rst line of this table the initial condition y 1 when x 0 determines the slope y x y 100 Since h 02 and y yo hfx0y0 the next value is given by 100 02100 120 This approximation is shifted to the y in the second line and the process is repeated to nd yz which turns out to be 148 In the table and most remaining examples we retain ve gures after the decimal point and the resulting approximate value of y1 is 297664 The exact value found from 4 is 343656 so the error is about 13 percent If we carry out a similar calculation with h 01 then the resulting approximation for y1 is 318748 and the error is reduced to about 7 percent roughly half of what it was in the rst instance Table 2 TABLE 1 Tabulated values for exact and nume rical solutions to 3 with h 02 x y Exact En o 00 100000 100000 00 02 120000 124281 34 04 148000 158365 65 06 185600 204424 92 08 234720 265108 l15 10 297664 343656 134 NUMERICAL METHODS 563 possesses two equilibrium solutions 17 0 which is unstable and 2 1 which is stable With the initial condition y0 01 predict what should happen to the solution Then with h 01 use the Euler method to march the solution out until x 3 What happens to the numerical solution 73 ERRORS The notion of error is of crucial importance in the study of numerical methods and we will give the idea some special consideration here We mentioned in the previous section that reducing the step size in the Euler method can be very costly This occurs for two reasons First the number of computations is directly proportional to the number of steps taken Thus raising the accuracy raises the computational cost Secondly a phenomenon known as roundoff error can become important This is a result of any computer s ability to represent only a nite subset of rational numbers Example Consider the benchmark problem 3 Let us examine what happens if h is made too small Let us suppose that our calculator has nine decimal digits of precision Let h 10 39 a very small step size that would seem to yield very accurate answers Applying the Euler method and computing the rst step we nd that the calculator obtains y y hfxy 1 10 39quot 1 10 The last equality in 10 is not a misprint Because of its limited precision ability the calculator represents y as exactly 1 Unfortunately the same thing will happen to y as well In this instance the Euler method would predict a constant solution to the test problem and roundoff error has produced a numerical disaster A detailed analysis of roundoff error is beyond the scope of this textquot As a result we will concentrate exclusively on discretization error in the rest of this chapter assuming that roundoff error is always negligible7 The local discretization error at the nth step is de ned to be 6 yx y This assumes that y is exactly correct As shown in the previous section for the Euler method this quantity is given by y Eh2 5k 2 11 But see Chapter 1 of R L Burden and J D Faires Numerical Analysis 4th ed PWSKent Boston 1989 for a very thorough discussion 7 Caveat computer p eissseeeTisLeee39eTieHs where r i t First rrete theft err the iTt39llEI39V39 i 1f39 tit 1 he x the quienttity y rs is iJ lJl1lEiEd by e pesaitfwe eenstent which is iedtecptendertt ef H Iei ttfhi Redtueirig the step size shy a faster ei redmgeezs the errer he39urid en the ieesi diseretieetiierr erreitr by a faster ef 6 the essmpie Ut1fDrtUtt1HtEit39 the stery is e hit mere eer epiieeted theta this since there is nethieg te prevent these lees errers Efrem atquot 1iJ39 5tLti iit tg es mengr steps ere tehen leads the the rretien et tetei tisiseretisetiert errer st the nth step En Te ES iiiTlE39tE this q esn tity rrete thet es the nuxmrerieei sehitien is merte39hed freer I te it in steps ere tekxerr end rt A5 ttnJ ih Asstuming the werst erase that is that ieeei errers isiiweys seed tegether teed rrever eerteel e heiuristie heur1d fer the tetei erret terse he ehteirreti MhE 07K 8 In Iii Eni E as Se fer the Euler rrrethed the ttetei eiseretisstien errer is rretser greater then semae eternsterrt times the step size Te iiiitt IIEIE thetse id ess let its eistirnete the diseretiesgtxien errers essreeieted the herrehmsrt tprteiiem 3 First riete that y 39 2 i2equotquot it is easy he see that en D 2 2 he 1 this qtreintiity assemees its iergest saline at sf 5 P Thus s Ir E ehi The tetei errer is heuhdee as well with Ei 0e Referririg te iTehie i in Seetierr Pd with it 7 the tetei rriliiseretieetieri errer at 3 1 is retmdeel te tee deeiimeii piteeets The errer heurrd is 2 H534 and es BEEPE1t2 iEri the tetei errer is iess then the heurid With it 0 the epprre prie te numbers eee be ehtsiniee frerri Tehie in StfC39tit1 tI I Q The tetei errtr is rt39 J25 whtiiei the errer heund is quotA We eise this seetien with tsemae prreetieei erhrise Sinee in mien prehirerrnrs ei eeirneer n the esieet s ehrtien is net eveirlehie fer gt2 iCilJi3E39i g err errer herinti eees ene knew when is tsmeii erretigh39quot Ones way used in preet iee is te esieeiete the numerieei seiuitti en seirersji times ELICCeS5i VEi hehing the step sis e W Wherr the 1 esttiits he Llerrger reherige withirt the preeisieru desired it W is gene but set irnfieliirhie bet that h rstmaii ieneugh By tihe same teiazen thew een ene eheeit E13 see whether it is tees A thet that t39 Ltttdeff erter is net er eeping inte the rprehilem One teefhrrique is the repeat 3 eraieuietiren resin etreedee39 JDFE39i5 itJ ft erithimetie Meet piregremhiirrg isrigrreges she trrrest eemputers steppert this eepehiiiity W herr reeeleuietee with estierrdee t39et i5 i39DI1 it the mimeri esJI reseits eihsrnge in any EL1iZ5ft I1 tit tii way it is simest a sure thing that serieus rrezeridaeh errers ere eeeertring Nevertheiests this test is net feeipreef fer it is aiiwteyst pessihie that the errers eriii net he eitsihiy rriernit ested even set esttended prieeiszierr NE hquot lquot f erget that as pewterhti as eempetters end rmrrieriesi methmis are they must be used with eejre NUMERICAL METHODS 565 PROBLEMS For the following problems use the exact solution together with step sizes 1 02 and 01 to estimate the total discretization error that occurs with the Euler method at x 1 1 y 2x 2y y0 1 y 1yy0 1 y e y0 0 y y sinxy0 1 y xy 12y00 Consider the problem y sin 310 with y0 0 Determine the exact solution and sketch the graph on the interval 0 s x lt 1 Use the Euler method with h 02 and h 01 and sketch those results on the same axes Discuss Now use the results in this section to calculate a step size suf cient to guarantee a total error of 001 at x 1 Apply the Euler method with this step size and compare with the exact solution Why is this step size so small GNU80DIN 74 AN IMPROVEMENT TO EULER Errors of this magnitude 13 and 7 percent are obviously unsatisfactory They can be reduced considerably by using much smaller values of h but this can have its hazards as discussed in Section 73 and a better approach is to develop more accurate methods For example it is not unreasonable to expect an improvement if we approximate the integrand 5 by the average of its values at the left and right endpoints of the interval that is by fxDy0 f xyx1 This is equivalent to using the trapezoidal rule for approximating the de nite integral in 5 Making the substitu tion we get y1 Y0 gquotfxo o fx1yx1 12 The dif culty with 12 is that yx is unknown However if we replace yx1 by its approximate value as found by the simpler Euler method which we denote by 21 yo hf xmyo then 12 assumes the usable form x Y1 Y0 g39ifx0y0 fx1Z1i 13 More generally I Y 1 yn 2fxk9yC fxk1Zk1 where Zk1 k hfxkyk39 15 TABLE 1 T abulated values for exact and nume rical solutions to 3 with h 02 us ing the improved Euler method x y Exact En o 00 100000 100000 000 02 124000 124281 023 04 157680 158365 043 06 203170 204424 061 08 263067 265108 077 10 340542 343656 091 NUMERICAL METHODS 567 Table 1 shows the approximate values of the solution obtained at the points x 02 04 08 and 10 by continuing this process The resulting approximate value for y1 is 340542 The error with this method is therefore about 1 percent which is a substantial improvement over the result obtained with the Euler method and the same step size With a smaller step size results are even better Table 2 displays the results of applying the improved Euler method to 3 using a step size of h 01 The relative error at x 10 has been decreased to about 02 percent roughly a fourth of that found previously Since the total discretization error is proportional 22 halving the step size leads to the result indicated above Clearly there is a substantial improvement in the accuracy of the improved Euler method at a rather modest increase in the complexity of the formula Suppose however that even more accuracy is desired TABLE 2 Tabulated values for exact and nume rical solutions to 3 with h 01 us ing the improved Euler method r y Exact Equot 00 100000 100000 00 01 111000 111034 00 02 124205 124281 01 03 139847 139972 01 04 158180 158365 01 05 179489 179744 01 06 204086 204424 02 07 232315 232751 02 08 264558 265108 02 09 301236 301921 02 10 342816 343656 02 NUMERlCAL METHODS 569 Obviously better accuracy can be obtained by retaining more terms in the Taylor series see Problem 8 The drawback to this approach comes from the need to evaluate higherorder derivatives of fxy These derivatives can become unwieldy in a hurry slowing down the calculation time for a given problem signi cantly Even more fxy may not be available in analytical form For example it could consist of discrete experimental data or itself might be the result of a numerical computa tion As such higher order derivative calculations are likely to be so inaccurate as to nullify any gain that might exist in principle Thus multiterm Taylor methods are seldom used in practice There exist much better ways to gain the accuracy needed with far less computational cost as will be discussed in the next section PROBLEMS For the following problems use the improved Euler method with I1 01 005 and 001 to estimate the solution at x 1 Compare your results to the exact solution and the results obtained with the Euler method in Section 72 y 2x 2y y0 1 1 y 1y y0 1 e y0 O y sin 15 y0 1 y x y 12y0 0 Think of some examples for which the threeterm Taylor method might work better than the improved Euler method In each instance describe why and if possible use a computer or calculator to illustrate the problem 7 Think of some examples for which the threeterm Taylor method might work poorly In each instance describe the source of difficulty If possible use a computer or calculator to illustrate the problem 8 Derive an expression for the four term Taylor method Apply it to the benchmark problem 3 with a step size of h 01 and calculate the solution out to x 1 Is any accuracy gained over the threeterm Taylor method Equotquot quot 5quotquot lt lt ill 75 HIGHER ORDER METHODS As with the improved Euler methods discussed in Section 74 the RungeKuttai methods can be derived from 5 by using a different 9Carl Runge l856 l927 was professor of applied mathematics at Gottingen from 1904 to 1925 He is known for his work on the Zeeman effect and for his discovery of a theorem that foreshadowed the famous Thue Siegel Roth theorem in Diophantine equations He also taught Hilbert to ski M W Kutta 18671944 another German applied mathe matician is remembered for his contribution to the Kutta Joukowski theory of airfoil lift in aerodynamics STD E mFFEaE rrmAL EGUATI0H5 aApprmimatimn for the interal us cnsider Simpm1 5 rule In this instance I bi 43u39nu39 439 7 539iJquotI 6 mym 3 where 321 2 13 2 0 rigur0us derivatim1 mf the fgurth nrdar Riunge K39utta method is byon c the scvnpc crf this chapter Rather vthan 539impEy state the resuMs we give h are3 an inuirE4uAe develpVmeunt nf thtis exIiremEly39 im pm39tant scheme far SQEwing nrdinar39gt di EslquotE Ii3J Eq4uatiuns In 0 the same way as we 3P PlJLiE39d the mhEr inVlegrati n fVc1rmu4Ias we must make estiimates of bath Jim and y1 The rs cifstixmate nf y E is nbrtained fmm E39uIer 5 methDid m1 If 53913 wlmrew H1 1 E hf 1J ya The farmr39 off I2 is QHECEEEETF since the Step sizes fmm x to Tlt1T2 is M2 To orrecxt this estiVmat e nf yW we raLlru Iam it again in WE fun lVlnwing way ml F12 J 4 5 K where n w rm hfrn 4 Vhf2 y mf2 Nnrwi to predict y we 115 this lat teT estima1Ae Em yw and the Euler m hnd p p where new m 35 hwquot2 yr Finail we let 6 hfx 391 ya THE Rung Ku tta metlmd is then obtained fmm 5ub5ti39i L1t ing Each of lzhese estimattzs int 3 1 to crbtain H n 4 2m 39539 2W3 4 wxith all p msious methudsA P one can be extended to any numb zr anif mESh paints in the matura way At carzh step rst Ezmipute the four quot It is wmth nxmng that mrim than 2 f urquotth nardTer Runge Kuttaa crmuIa calm be 39i jEI iquotr39iiE SEE E CMnlh aIn L39u39lVher n d1 J On Wikes Apph7Enri39 Nurnerice M rI39md539 Wilma Lhl zw quotr39nrrk PP5 3639I 3i53 LFUE as shvIvr39L hut jnIIe5 ing hquotiiiftf39ri43l iZi3iiEE L55i l t DE Hwi puint NUMERICAL METHODS 571 numbers m m4 ml hfxk k h m quot72 391 hfxk 39i k l 31 h m3 hfxk 239yk l quot23 m4 hfxk h yk m3 Then yk is given by k1 k l P 2m2 l 27713 quot4 22 This powerful method is capable of giving accurate results without taking h so small that computational labor becomes excessive or that numerical roundoff becomes a serious problem The local truncation error is 6 y h5 180 where x S x 5 x and the total truncation error is proportional to h This is one reason for its remarkable accuracy We now apply 22 to approximate y1 in our benchmarks problem 3 With h 1 so that only a single step is required we have m2100S1052 m310051125 m4101125 45 so that y1145453417 This approximation is even better than the improved Euler method with h 02 In Table 1 we show the result of applying the Runge Kutta method to our benchmark problem with h 02 Note especially that TABLE 1 Tabulated values for exact and numerical solutions to 3 with h 02 using the RungeKutta method 1 y Exact Ii o 00 100000 100000 000000 02 1 24280 1 2428l 00044 04 158364 158365 000085 06 204421 204424 000125 08 265104 265108 000152 10 4 343650 343656 000179 NUMERICAL METHODS 573 76 SYSTEMS Heretofore our numerical methods have been employed against rst order initialvalue problems It should be clear that many important physical problems are modeled by second and higher order equations such as vibrating mechanical systems or even directly as systems of equations such as predator prey systems It is therefore natural to seek ways in which our methods can be extended to treat these types of problems Since dzydtz f t ydy dt can be transformed into the system of rst order equations dydt x and dxdt f t yx it is customary to transform all higher order di erential equations into systems of rst order equations In this section we will discuss formulas that explicitly treat systems of two rst order equations but the results can be generalized to more equations with relative ease It should be noted that serious scienti c and engineering applications employing models composed of complicated systems of differential equations are almost always solved with methods albeit with a bit more sophistication very much like the ones we will describe here Our objective is to formulate methods for generating numerical solutions to the following system of equations x f my 23 y gtxy 24 with initial conditions X00 350 YOU o 25 We assume of course that the functions f and g are sufficiently smooth so that unique solutions to 24 24 and 25 existquot As in the previous sections we seek to construct approximate solutions x and y to the system at the points t to t to h t to nh The Euler method takes on an entirely analogous form for this case and is given below xk1 xk hftkXk 1 26 k1 k h8 kxk k 27 where k 0 1 n 1 The expression for the local truncation error is more complicated for the Euler method in this instance but it remains true that the total discretization error is proportional to h quot See Chapter 11 574 mEFEmHTmL EQUqI1DHS C3unside r the fnllowing lVimear sagend urder nonIhumngenenus di quoter Ential eqlmtinnz M t3 P 432 CD51 E at L with initial cnndiIinn5yAU t yU P u Equaitinn K can be thugm of as a model Em an Aundamped 5pring Vmass sub ject to a sAinusuidaVI Emrerinr dIiwin fsJrcampe tim I U Eh Amass lies at its equilib1riuvm pnsitlii n with n initi a valmzitjy The exact sulu un In 2 0 x 23 0Cr msi CD5 t C at intn sjysrc Em fmm we rst lat m L Then 1 a4Ay ms t g quot39 I n with initial E diIiiUHS Ji U yU L i Table 1 csnntainse the t1abulated results gr this sgwstvsm an thae intefrvai U V I 5 1 using the Euler method Wm i 011 Oh that tha 1quotEJT t i E cvnm Emquot y starts gut EJE39I1TBmE39i1 large dE ElfEaSE5 EU a rathar small value and then begins tag i nc1 eaasc agai11 See Prmblem 5 far 3 discussiuIVn Bf this phenm em nn TalnuLlaled values rfm exact and mlmerical sr lVu nns In 29 and v with F 1 using the Euler me4thud 33 IE Ky E iflmf y E36 1 49 CW 11 02 U E9339 03 04 5 i6 39l39r39 k 1 I 3393 M 1 44545 4Pfd EU 5 232 Z Slliltlai U39A92E f ISL 43Tl39D r 593U39 ULEIWI ISL 155 V 39l9 399 Du 24302 D 5 295 U6 El R R 23 DEM43 05111 U43l39i15 DAE2E Uh33BTl 2 D325quot1 Bl 111243 0 P 53433 pR 19339 I124 19 iU39EB2quot 1v D 3 1 8 H2 31 21 14 8amp3 quot39 37 041 43 30 11 This mbLmlatiusn 5hnuld mnvincc anmnse Esfhhumd aurh c nvincing ha nmded 391quotjp ilI39 lg such 3 cai1cuEatin by ihand that thaere is tmIhing Eikuf 3 r39umffp utEL tuagether wifh E1 and prugr39armming anguagE fnr av cumpli5hir1g 39E L1392h a E3515 Imaxgme what39 it was WE3 in Izht mid rda339sa pmquotWrld War III when virIuaMy all engineEr ing rtJmfpmaIiuns wvzrr dzquotne w39iM1 a rpermil pa IEI and perihap 3 desk calcu latu39r NUMERICAL METHODS The RungeKutta method for this system is where Xc1 Xk 4 1uk1 I12 Hk3 W4 k1 Yk l Vlt1 Vlt2 Vlt3 3939 Vk4 Mlt1 hftkxkryk Vki 1 hgtkvxkrykr h 1 v k2hftk 1xk392l l9yk Vk2hgtk xkE39 yk39S39 2 2 2 0 hf 5 x 4953 y 3 k3 k 2 k 2 k 2 Vk1hgtRf1xkE grykZ2 2 2 2 M4 hftk h X7 W3 k Vlt3 Vk4 1780 4quot hi Xc k3 k Vk339 575 31 32 33 The total discretization error for this more general Runge Kutta method remains proportional to h The numerical solution of 29 and 30 with a step size of h 01 is displayed in Table 2 Note that the relative error is signi cantly smaller than that seen with the Euler method as shown in Table 1 and furthermore the relative error does not exhibit the same degree of uctuation as that case TABLE 2 Tabulated values for exact and numerical solutions to 29 and 30 with h 01 using the RungeKutta method 1 x y Exact x Exact y E for y 6 00 000000 000000 000000 000000 01 009917 Q0O498 009917 000498 00006 02 019339 001967 019339 001967 00018 03 027792 004333 027792 004333 00022 04 034843 007478 034843 007478 00023 05 040117 011242 040117 011243 00024 06 043314 015432 043315 015433 00024 07 044223 019829 044223 019829 00024 08 042726 024196 042726 024197 00023 09 038813 028293 038812 028294 00022 10 032571 031881 032571 00021 031882 pUz EIIFFEEENquotFlJALEIE1iMgeTlUH 1 z x ethee Euler emetlhed with step eizise eh 2 z te e1raueeIe the eemtien m yquot a fl yf U1 39391 0 at I 072 and I Q Cempare yeur reeu39te tee the vezmet seletion Use the Euler methed quotwith step size UL tje evaluate the seluertei en tn the feewing s39stem ef eqgueeatriene at I E 05 I E ya 1 E I39ll with xEl x M0 1 Use the Rungee I39uItae msetheed and 3 eemputer te evaiuatee me eeJm39ie en quotte y yIA e x I z E1 y 1 and y TJ 11 at t r Use Ste sifzee ef j L2 and 0 t Genereiiee the fe139meiletien ef la he Euler methed In a esystem eff three 39 r3t order erdinaw dei ereential equaIiensee 9 Using the results lietecl in Table ekeetehe the graph ef 35 and i venue r eEIpe1eaien the 1immvl i39Bt DlrI in the reletiave errel Dees the same error fbe I1evier39 eeeur fer x and 1 Why dees the HunVgee Kuttea errr seeee Tame 2 net behave nlhis we NUMERICAL TABLES Table I Trigonometric functions ANGLE ANGLE DEGREE RADIAN SINE COSINE TANGENT DEGREE RADIAN SINE COSINE TANGENT 039 0000 0000 1000 0000 1 0017 0017 1000 0017 4639 0803 0719 0695 1036 239 0035 0035 0999 0035 4739 0820 0731 0682 1072 339 0052 0052 0999 0052 4839 0838 0743 0669 11 1 1 439 0070 0070 0998 0070 4939 0855 0755 0656 1150 5 0087 0087 0996 0087 5039 0873 0766 0643 1192 639 0105 0105 0995 0105 5139 0890 0777 0629 1235 739 0122 0122 0993 0123 1 5239 0908 0788 0616 1280 839 0140 0139 0990 0141 5339 0925 0799 0602 1327 939 0157 0156 0988 0158 5439 0942 0809 0588 1376 1039 0175 0174 0985 0176 5539 0960 0819 0574 1428 1139 0192 0191 0982 0194 5639 0977 0829 0559 1483 1239 0209 0208 0978 0213 57 39 0995 0839 0545 1540 1339 0227 0225 0974 0231 5839 1012 0848 0530 1600 1439 0244 0242 0970 0249 5939 1030 0857 0515 6 1664 1539 0262 0259 0966 0268 6039 1047 0866 0500 1732 1639 0279 0276 0961 0287 6139 1065 0875 0485 1804 1739 0297 0292 0956 0306 6239 1082 0883 0469 1881 1839 0314 0309 0951 0325 6339 1100 0891 0454 1963 1939 0332 0326 0946 0344 6439 11 17 0899 0438 2050 2039 0349 0342 0940 0364 6539 1134 0906 0423 2145 21 39 0367 0358 0934 0384 6639 1152 0914 0407 2246 2239 0384 0375 0927 0404 6739 1169 0921 0391 2356 2339 0401 0391 0921 0424 6839 1187 0927 0375 2475 2439 0419 0407 0914 0445 6939 1204 0934 0358 2605 2539 0436 0423 0906 0466 7039 1222 0940 0342 2748 2639 0454 0438 0899 0488 71 39 1239 0946 0326 2904 2739 0471 0454 0891 0510 7239 1257 0951 0309 3078 2839 0489 0469 0883 0532 73 39 1274 0956 0292 3271 2939 0506 0485 0875 0554 7439 1292 0961 0276 3487 3039 0524 0500 0866 0577 3 7539 1309 0966 0259 3732 3139 0541 0515 0857 0601 76quot 1326 0970 0242 4011 3239 0559 0530 0848 0625 77 39 1344 0974 0225 4332 3339 0576 0545 0839 0649 7839 1361 0978 0208 4705 3439 7 0593 0559 0829 0675 7939 1379 0982 0191 5145 3539 0611 0574 0819 0700 8039 1396 0985 0174 5671 36 0628 0588 0809 0727 8139 1414 0988 0156 6314 3739 0646 0602 0799 0754 82quot 1431 0990 0139 7115 3839 0663 0616 0788 0781 8339 1449 0993 0122 8144 3939 0681 0629 0777 0810 8439 1466 0995 0105 9514 40quot 0698 0643 0766 0839 85 39 1484 0996 0087 1 143 4139 0716 0656 0755 0869 8639 1501 0998 0070 1430 4239 0733 0669 0743 0900 8739 1518 0999 0052 1908 43 0750 0682 0731 0933 8839 1536 0999 0035 2864 4439 0768 0695 0719 0966 8939 1553 1000 0017 5729 4539 0785 0707 j 0707 1000 90 1571 1000 0000 577 STE DkFE EHT1AL EEFU ATI39DH Tabla 0 E3IiWI ElaEn39Ii I l mtii x ai F j H i 3 i 1 1 T A7 urgt39 T 7 mm A E15 3 Lquotil 5 E I Z I EASi dJ L Q In Z M943 M a4aJaquotu 1 amp392 ms 113591 23 I 5445 uma 3 v lj 4 anar 29 mm uum tn2s A aM uma 10 2n naa A nuf49 a l13 I ll 3amp1 E N03 3 I 22 WE ti ua35 mm A mm 32 2a533 n nr4u3 049 wanna A nma 2 mt Iur3amp9 ms 15m W ums 2 E39 4 A uni34 EI5iEl39 um s1 mamas i 33 13 15 00m 1I5 5 mm msm 36 Jules n H3 w ua22 1 nau a 3quot 344 n147 Ms Lam 39D1lTEI39 33 44lt39nm nu24 nimr 2m3s vu4955 39 4902 002m EI 5 7 2IIm ncm1 am 541593quot nm33 rIIEII 1 39255 II4 iI339 4 I w34n 023A 139 E6 035 2339amp wm14 42 us ampampamp uI15u 1 091 F n4maamp 43 13100 nm3si ms T T a45 m uvm23 Lu am aam 45 uen uu1 I M 45 9943 nmm i 12 332m Mm w 47 1 amp9 5 EM1EI9 L3 3 FJ3939i mm 43 ll2ii5 4 ma2 14 40552 u24m 49 m29 J mim 4431 mm A 5 5 1458 agmmr L45 49530 MEI 19 5 40343 WISES 113 1s3 m El 1 51 1 m5gts5 W L LE 53 um53 A 5 zaarm L9 359 l i439i amp 9 mm am 1339 1 am am 391 i l 21 31652 tn15 1 A 12 amsu j 1 ms 39 I I 323 M 39U 39939 3939 I t mm i2a27 I mmu NUMERICAL TABLES 579 Table 3 Natural logarithms In x log x This table contalns logarithms of numbers from 1 to 10 to the base an To obtain the natural logarithms of other numbers use the formulas In 10 x1nx In 10 In X In t39 In 1039 10 In 10 2302585 In 102 4605170 In 10 6907755 In 10 92I0340 In 10 11512925 In 10 13815511 1 o I 1 2 3 4 5 6 7 3 9 10 000000 0995 19180 2956 3922 4879 5827 6766 7696 8618 11 00 9531 0436 I333 2222 3103 3976 4842 S700 6551 7395 12 01 8232 9062 9885 0701 1511 2314 3I 11 3902 4686 5464 13 02 6236 7003 7763 8518 9267 0010 0748 1481 2208 2930 14 03 3647 4359 5066 5767 6464 7156 7844 8526 9204 9878 15 04 0547 1211 1871 2527 3178 3825 4469 5108 5742 6373 16 04 7000 7623 8243 8858 9470 0078 0682 I282 1879 2473 17 05 3063 3649 4232 4812 5389 5962 6531 7098 7661 8222 18 05 8779 9333 9884 0432 0977 1519 2078 2594 3127 3658 19 06 4185 4710 5233 5752 6269 6783 7294 7803 8310 8813 20 069315 9813 0310 0804 I295 1784 2271 2755 3237 376 21 074194 4669 5142 5612 6081 6547 701 7473 7932 3390 22 078846 9299 9751 70200 0648 1093 I536 1978 392413 ms 23 03329 3725 4157 4587 5015 5442 S866 6289 6710 7129 24 08 7547 7963 8377 8789 9200 9609 00I6 0422 0826 I228 25 09 1629 2028 2426 2822 3216 3609 4001 4391 4779 5166 26 09 5551 5935 6317 6698 7078 7456 7833 8208 8582 8954 27 09 9325 9695 0063 O430 0796 I160 1523 1885 2245 2604 28 10 2962 3318 3674 4028 4380 4732 5082 5431 5779 6126 29 10 6471 6815 7158 7500 7841 8181 8519 8856 9192 9527 30 109861 quot0194 0526 390856 I186 I514 I841 2168 2493 2817 31 11 3140 3462 3783 4103 4422 4740 5057 5373 5688 6002 32 11 6315 6627 6938 7248 7557 7865 8173 8479 8784 9089 33 11 9392 9695 9996 0297 D597 0896 391 194 1491 1788 2083 34 12 2378 2671 2964 3256 3547 3837 4127 4415 4703 4990 35 12 5276 5562 5846 6130 6413 6695 6976 7257 7536 7815 36 12 8093 8371 8647 8923 9198 9473 9746 0019 0291 0563 37 13 0833 1103 1372 1641 1909 2176 2442 2708 2972 3237 38 13 3500 3763 4025 4286 4547 4807 5067 5325 5584 5841 39 136098 6354 6609 6864 7118 7372 7624 7877 8128 8379 40 138629 8879 9128 9377 9624 9872 OMB 0364 0610 0854 41 14 1099 1342 1585 1828 2070 2311 2552 2792 3031 3270 42 14 3508 3746 3984 4220 4456 4692 4927 5161 5395 5629 43 14 5862 6094 6326 6557 6787 7018 7247 7476 7705 7933 44 14 8160 8387 8614 8840 9065 9290 9515 9739 9962 D185 45 15 0408 0630 0851 1072 1293 1513 I732 1951 2170 2388 46 15 2606 2823 3039 3256 3471 3687 3902 4116 4330 4543 47 15 4756 4969 5181 5393 5604 5814 6025 6235 6444 6653 48 156862 7070 7277 7485 7691 7898 8104 8309 8515 8719 49 158924 9127 9331 9534 9737 9939 0141 0342 0543 0744 50 160944 1144 1343 1542 1741 1939 2137 2334 2531 2728 x 0 I 2 3 4 5 6 7 8 9 Note The quot indicates that the rst two digits are those at the beginning 0139 the next row 530 DVNFFEREI4mA39L Enumtums T3rlv Haliur hEgariIl139r In J lvm I CmInr 1 J I 5 a m n g I l L si l l5 Tl I46 HEti IQTEQTE LTEETE LT4 iT a 5T3amp LT 95 Il i lli susa 959 3325 mass 2455 222 5953 wags 13a 3315 525 Id EDIE H333 2amp1 439 ll 13 1542 35 544 E335 9E4 I l EHII 5E l i amp E TdI SE11 T513 9313 IE9 IRES 4146 T HJ lwnaj v Ti 553 mu 95 njan Jlamp 491 aaaq HA1 9 Ii 1342 SD94 EEEE 35H EH13 95 1I1I 2lE EEWJ 33 Dill IQWE EEF5 54 l SE42 i aamp1 Huang 6551 Eamp55 E IE 5 IJ JEE 39 Q jwa1 F313 smw 945539 m uiif39 vnsaa tueas I 3 3129 IE245 ALEHMS 1sg5a1 LEE i L9 Ei1 L9ampampE39 L 1332 i9459 I 1 991 game 4214 5155 T314 asja l I339 32 iavaq hligz E d32 5942 dE a l BSD 1 E 344 431 x EWJE 453a T143 335 553 6351 19 95 229 Jrjn I i l mauh Jams j i ia s 35 wad 5403 794 095 Edl J ia 31 S443 ll l E5 E 452 9El2 q m 1miEf 3 531 EETE B433 9912 J E 2amp5 isms ns El n 359 5amp7 wmaa E555 E IUD 4 5i 5s 7A 139lT9 Entjisi fLd E taaT l Ei aa n Eill LEENS 21 iIE Vi Ii T is 35 592 233 E ii 294 1152 354 ll 391 63 051 Gina EFEF JETS H35 i EH39 a lh 5T9 I B5 i9amp1 H334 UHET 2023 Jl A d 593a 9t 1 Ti aasm l 3213 9ain gang 032 BEBE Zl i 2231 343a l TEE I443 Jnnm wi 439 bIT9 m3 ESE 93 I323 E l JEEE Eli 6433 T l Fi E 115 i 2amp3 E 5234 560 FEE i ll TIquot39Mia EEG EI94 a1uaW 3 EQ43 356 3 9l T EH39 9n E SJ E 33 3amp7 1sns 2J diJ jEJl E 2J3 1J E145 935 515 14amp 29 4134 5292 gas 1539 E11 943139 Has w IE5 gig 439 E E339 QESE 119 l 3l3 d35 552 saw H1 a 2 7 a aan Jl 3393 Jai S d man 1 i l i ns i 162 2225 341 R l 59 41 ETFamp S il 995 T l 33 E135 Qi 9 4 I4l 314 quot 91 K 39 3 wamp9 291 I334 3535 1 4941 hi l T245 Eii i E 4 A Q lw 4 355 naa 2 39 lEF ma 2amp2 3539 Q47 1quotla Egan 115 3143 354 4i 4i 213 321 l d i l w Taaw 3amp4 455 ii i 134 31242 41993 544 4BE I52 a54M ii n 2354 343i mli IE5 253a E 3333 645 53 EQTHT 553 5 39E 9 T9 T39 i19 1d E d 9 5E Tm5 3 Ei 3353 i S i EH34 Q m 31m 313a 152 i 353 ai n i 35 5 a I NUMERICAL TABLES 531 Table 4 Common logarithms 1030 x x 0 1 2 3 4 5 6 7 8 9 10 0000 0043 0086 0128 0170 0212 0253 0294 0334 0374 11 0414 0453 0492 0531 0569 0607 0645 0682 0719 0755 12 0792 0828 0864 0899 0934 0969 1004 1038 1072 1106 13 1139 1173 1206 1239 1271 1303 1335 1367 1399 1430 14 1461 1492 1523 1553 1584 1614 1644 1673 1703 1732 15 1761 1790 1818 1847 1875 1903 1931 1959 1987 2014 16 2041 2068 2095 2122 2148 2175 2201 2227 2253 2279 17 2304 2330 2355 2380 2405 2430 2455 2480 2504 2529 18 2553 2577 2601 2625 2648 2672 2695 2718 2742 2765 20 3010 3032 3054 3075 3096 3118 3139 3160 3181 3201 25 3979 3997 4014 4031 4048 4065 4082 4099 4116 4133 26 4150 4166 4183 4200 4216 4232 4249 4265 4281 4298 27 4314 4330 4346 4362 4378 4393 4409 4425 4440 4456 23 4472 4487 4502 4518 4533 4548 4564 4579 4594 4609 29 4624 4639 4654 4669 4683 4698 4713 4728 4742 4757 30 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 31 4914 4928 4942 4955 4969 4983 4997 5011 5024 5038 F343 gt4 I eLeJ1JLaLnJuJleJLaJL3 gtLoJIJO VOOOIOUn Gskutv Ch LA to LII 1 Ox Ix La Ox Vs tx Us 3 xi LA in O U Ix LII Uu on be l O K Is UPI Ox xo 39aJ O O U N 0 J n O LII to ox co to A t Ox U N Ch 390 L A Us 0 U L O O s 1 O U Lu Ch 11 45 OO O la La Ch N l V O 6435 6444 6454 6464 6474 6484 6493 6503 6513 6522 46 6628 6637 6646 6656 6665 6675 6684 6693 6702 6712 47 6721 6730 6739 6749 6758 6767 6776 6785 6794 6803 48 6812 6821 6830 6839 6848 6857 6866 6875 6884 6893 49 6902 6911 6920 6928 6937 6946 6955 6964 6972 6981 50 6990 6998 7007 7016 7024 7033 7042 7050 7059 7067 51 7076 7084 7093 7101 7110 7118 7126 7135 7143 7152 7160 7168 7177 7185 7193 7202 7210 7218 7226 7235 7243 7251 7259 7267 7275 7284 7292 7300 7308 7316 7324 7332 7340 7348 7356 7364 7372 7380 7388 7396 7404 7412 7419 7427 7435 7443 7451 7459 7466 7474 7490 7497 7505 7513 7520 7528 7536 7543 7551 7559 7566 7574 7582 7589 7597 7604 7612 7619 7627 7634 7642 7649 7657 7664 7672 7679 7686 7694 7701 7709 7716 7723 7731 7738 7745 7752 7760 7767 7774 LhLlI1lLIh KIKALII OOOO1J1 51J1 xl D 1 Is LD1FFEFEHTL9L Eu399run999 1 5t e at Eumm am L3EI I l5 3911IifI5i iijlugm 1 IquotZ73999m J D I 2 9 4 4 3914 J a9w acht1 hDs F E rwvhaw as I9II4n2 9945 9951 9199 9 59 911 9293 939 I 5 9395 99 115 9999 39 19959 II 9 925 939294 93 9935 945 5 E15095 913 My 999 9595 96933 9939 9939 E 9 quotquotFquot 3 9amp9 3t 1 EVTQ I WEE 9332 T 9921 9965 9 E92 E1 V9STE 99 I T W il 195 El IBM La 9999 9999 9999 9599 9599 9999 9999 9999 9994 9 I2 9999 99 9999 9999 39T9r D99im9l painls are nmittrd in H3939 iah the mI39i99 Iii Ill 9919 9999 99l3 9959 9995 9999 9399quot 339 W835 i939quotT 9999 T9999 U939 99E9 1 339 FE 9139 9 7 99933 9399 9959 9999 999 93993 953 9 623 939 5 9999 99 L9 9959 9993 9999 T395 W J39 NET 395155 3 ti E2 3 B9 EEJIQ 33322 3945 9999 m91 that lngm I J3lt iUIEEZIIJ 191 mI DL II EJ IL I 39I3 9991 fgIquot39 JBIEEI I 1U EI5 199 Fnur9d9 ia 9199vpE99 am 9rac91 NUMERICAL TABLES 583 Table 5 Powers and roots 1 x J x 1 x x J r 01 1 1 1000 1 1000 51 2601 7141 132651 M 3708 2 4 1414 8 1260 52 2704 7211 1 140608 3733 3 9 1732 27 1442 53 2809 39 7280 148877 3756 4 16 2000 64 1587 54 2916 7348 157464 3780 5 25 2236 125 1710 55 3025 7416 166375 3803 6 36 2449 216 1817 56 3136 7483 175616 3826 7 49 2646 343 1913 57 3249 7550 185193 3849 8 64 2828 512 2000 58 3364 7616 195112 3871 9 81 3000 729 2080 59 3481 7681 1 205379 3893 10 100 3162 1000 2154 60 3600 7746 216000 3915 11 121 3317 1331 2224 61 3721 7810 226981 3936 12 144 3464 1728 2289 62 3844 7874 238328 3958 13 169 3606 2197 2351 63 3969 7937 250047 3979 14 196 3742 2744 2410 64 4096 8000 262144 4000 15 225 3873 3375 2466 65 4225 8062 274625 4021 16 256 4000 4096 2520 66 4356 8124 287496 4041 17 289 4123 4913 2571 67 4489 8185 300763 4062 18 324 4243 5832 2621 68 46214 8246 314432 4082 19 361 4359 6859 2668 69 4761 8307 328509 4102 20 400 4472 8000 2714 70 4900 8367 343000 4121 441 4583 9261 2759 71 5041 8426 357911 4141 434 4690 10648 2802 72 5184 8485 373248 4160 529 4796 12167 2844 73 5329 8544 389017 4179 576 4899 13824 2884 74 5476 8602 405224 4198 625 5000 15625 2924 5625 3 8660 421875 4217 676 5099 17576 2962 5776 8718 438976 4236 729 5196 19683 3000 5929 8775 456533 4254 784 5292 21952 3037 6084 8832 474552 4273 841 5385 24389 3072 6241 8888 493039 4291 900 5477 27000 3107 6400 8944 512000 4309 961 5568 29791 3141 6561 9000 531441 1327 1024 5657 32768 3175 6724 9055 551368 4344 1089 5745 35937 3208 6889 9110 571787 4362 1156 5831 39304 3240 7056 1 9165 592704 1 4380 1225 5916 42875 3271 7225 9220 614125 4397 1296 6000 46656 3302 7396 9274 636056 4414 1369 6083 50653 3332 7569 9327 658503 4431 1444 6164 54872 3362 7744 9381 681472 4448 1521 6245 59319 3391 7921 9434 704969 4465 1600 6325 64000 3420 8100 9487 729000 4481 1681 6403 68921 3448 8281 i 9539 753571 4498 1764 6481 74088 3476 8464 9592 778688 4514 1849 6557 79507 3503 8649 9644 804357 4531 1936 6633 85184 3530 8836 9695 830584 4547 2025 6708 91125 3557 9025 9747 857375 4563 2116 6782 97336 3583 9216 9798 884736 4579 2209 6856 103823 3609 9409 9849 912673 4595 2304 6928 110592 3634 9604 9899 941192 4610 2401 7000 1 17649 3659 9801 9950 970299 4626 2500 7071 125000 3684 10000 10000 1000000 4642 Lrb bJgttsI541135 nuuuwuwwmw wwwmwtummmru 0OO1C7UJLJtu OOoJOL1I LaJIJ ooooxx E coooooooo xoooooooaooooooaoooo ooqnI1 O cogtc7v41J OoooIovn gt4JN ooooa1 mF FERENTML EEQuampT1m45 quot uhluA E Facmr39iili H 31 H 1 vEuEE 39 3333 nnm39jnrr 4 Avmzmwa 2mn xu Er 30 Em E E39I EI rih Em E D l ma El3 Em Ll E92 A1 1 y Em 5m iiiu mu Em 403Z pQ EH4 3262E E w EM l i l I iilquot Eljg 399 Ii E 1 Y Xv EEI J 73 E E1 939 I 2130 i31ElErT 3630 E M 2 W39E F 39 E EE EH3 33955EETquot a1E ill VEM 45392i37 3iJ 539 E H15 Lili 4 TECIEMI E41 mwwmw Eurruuiu j has ru D P91 l 39vJ Ii a 2413290 EDEIEE En E 51 9 9 2112 Em im D mE E2 255533 15739 E22 ml el s399HD13i cr I J5 E I 2 Lin T E25 mu 329i am I 3 Elm LQEBEE am E23 3U 3E3 344m E29 3S1iHi l9E1 E3r o 3593 EM 337223 E E542 E33 263 I 3D E 3E93 E35 339 E3 3l FrEiI SE E3E 295233 13999 H E33 1H k 1 gh iuumaynquotu En wnzaanaaan mfhei391393quotl H39iquotquot7quotquot5quot39quotquot39equotquot l3quot4iquotquot39 3 Jfllil I 4 W645 Et1 i 3l 9quot395393 3E a939 E41 I339 I53 EJDQI E43 5 E2EEEE 51n1139 Edid IE39L 3EquotT3 E l E il ia E IE 5 I is 2313525 E4quot A 3345 515 E I 3 EMS l45UL39 61 HE E51 23513I4l r Ea3 I53 fE5J Z39E39 53E SHE E561 lhm oll EEEIE EH 5 521 2 ii 593 ET5 Z5iEEiI23 E4 M II E59 li24 an 3939 l5E39l393J7 I E39 EE3 1 EEHJJEA E i A ll I2dt i 939 EUR EM NJMi H39aIus arc gimrn i rm I fienali t 1m39Iaiv 1i wMh uh uxnem ermled quotI13 E Eur mampr 265252 SE93 E33 dEI39i rs 2 5E5EHEW I 331 NEW ANSWERS SECTION 2 p 9 1 1 2 a y e3 x2c d yxsinquotxV1x2c b ylogxc e yx l0g1xc 1 1 C y equot2c f y log1x2c g y log 3tanquot39 P H 1 c 1 h y tan x2 c n y log cscx cotx c i x cy 1e 0 y ccosx J x ff c P y csecx k siny cxe 2 C1 Y lcgci I y C6 r y 6 m x3 3c0sy c 585 E 3UATI H V L 0 3 W12 E xe i 6 3 d y EngI J I 1 A V 1T 3 Ifb saru3r 11 E Iugi T 1 22 P 1 W4 gr I If 3 3 33 2 25quot quot1 L1 sin was A2 1 b I 1 lag I E 2 I e sinx 05331 tam 1 x p Equot I lugxhr 2 y J 3 m Eli IE2 2v tE s l39jLE39 39h Ii iIngt lFc2 l p ta39nquot r M M M M 1 H p 51 E c r 39l ms 9 b 9 233 cg Vd A3 21 c xi 4 El 12 riyl I I Time orthaganal traVje Iories are ampii5E5 and are mn m and Amre e1n ngat eampd the J diIEcti n as n taken t be ilargcr and larger r V Ea 215 65 4 r CH1 C205 6 5 y Zxy ue fammhf 15 Jeffa r39rh g nmquot run the 5cn5Elt that when a cunre in the fammr imer5EaIt5 ann therA ru4rwe in the f39amiI439 it is rtl7mgm1aI to it 6 a II39 E xi p wzti b J E ff ya if pn u t 2 c39 39 rfgjlr H I or r 2 Resin E Id i H U1 3 H m r ruse 39 1 3 11 xy Equot T1einttr39sectiUnjs of H1E c rindler5 xy 5 c with t h E sa dd1e surface 2 E 251 10 Kai W xiv E H d 1 W If Eh I2 ya W 5 ilxyi E U Iiyquot2 E W E I yp l 3 E I vyA II1 ll SECHN 4 p W0 lung 2 gay T E rE T395 b about pe rxccnL 3 a A Dfiki 1 c mm in 3 40 SO 10 7 T5535 16 ANSWERS 537 W 1 A P 1 39 a k k e c T klogw Woyears b W kP d about 1386 years If x xt is his wealth at time t and t 0 one year ago then x 202 t Thus in 6 months x 40 million dollars and at the end of 1 year as t gt 2 x becomes in nite At about 1011 PM 3531 In the year AD 2076 66 billion b About 152 grams x x39 39 when x 1x X0 X quotquot39 xCkxquot 2 I About 3535 percent about 3125 percent About 133 days About 1353 percent If B A then x kA2abt kAabt 1 and if B lt A then i ek A Bab A Be kABabr The rst formula is the limit of the second formula as B A students should prove this by using l Hospital s rule 17 1 Ax 1 Ax39quot39 39 1 Ax 1 Axo 39 x0 18 x x9 1 xequotquot 19 40 log 2 E 2772 minutes 20 2log2 E 139 hours 21 No later than 36 minutes after the smoking starts 22 40 feet 7 9 3 Nil d I 23 251quot aquot 6251quot quot 5 l 25 3g 5 lhours log2 26 60 27 16 28 At 6AM a About 3330 years 1380 BLc C about 10510 years b about 3850 years 1900 Bc d about 7010 years DWFEHENTML E U TICl39NS 5 p Y j II E M 1 E 1 E121v39aju f f W f fE E39l39 when I Q1 139 R Awhen r vi g he V wi139ich is app rir1 1imateIy P S misr395ecun d1w Et 2 EJr J Aquot pcG T the t enn 39maV vell city is Q R MISCELLANEGUS PRELEMS FCIR CHAPTER 1 p 44 K 2 M 4 The intVersectim15z of lhe c3rlinders1f rg uyquotquot Wi th 4x1 y 42 FL FL 14 M FL J p H 1 hours bve39EI1irc nmim r M quot2 IN D116 mnrc mmth After l Ig2 min4ute5 1DDRf e 1 nnhiute I 1 fM vsTc nndsw 145rW Thc 5h1133 Hf urns 5 urfac nbtaintzd by rEmluing y I aharJutrtl1 e y arzi5 25h Elngfd UC se znndsm if E r 2 rm The Pre5idnt uT 0 E Ey R miii GU 6 miles mw ard the mrigin an t4h en mm nu tward ung mm of the 539piraIs H F E1 quotr3 Q T TE H mall dtsmzainc 11 9 7 I1 49 L x J J ag IKUJg2cV fl 1rJEr m E Jr Ex A s8 3 ya 13 Ef W F7 rxlix quot 39 l39f C y x E EIquot1392 5 cn s r T lng I D quotM J 9 1 Il giv quot y cxEf l mf A11 33 1 IUv I39 yfv rrxzg m y I ling ncz 2 322 4 3 J y V tan 1 5 VB j eyu tan E y J 2 A a39 11 a tan 2 logx 12 y 52 c byx 5logxy 1c c logy x x D2 2tan 39 c d x 2yx 2y 4 c e 2 y 3 cx 1 a n 12 x cequot 2 b n 34 2 Sxyz cxm c n 1 x eyequot Ea r ce Sin polar coordinates b r cc c x2 y2 c SECTION 3 p53 11 13 14 15 16 17 18 19 yy2x cxyorxy2 x2cxy 21 22 quot quotquotquot SquotE 39quot xy log y2 c Not exact 4xy x y c Not exact 39 xy sinxy c Not exact xe sinx cosy c H 5393 x cos c or Y lt15lt Not exact 10 x2y3 y sinx c 1xy logx y 2x C x2yquot xsiny c log1 xy x2 c 1 xy 3x2 2x2 y c Not exact xeyz cscy cotx c x yzcoszx c x2 y2 C2 x31 logy y2 c x2y24y2 x2 c a n 3 xzyz 2x3y c bn1x2e2quot c ANSWERS 589 nmFFEmErrnnL EI39Zl U AT I HE z r quot 1 A2 2 1 2 3 P1 f I cw J W 1 N 2rzv 1039 A2 p M 5 0 F Y CI 0 1 d u sizny as sin y y E rE Va requot e sin y E c c M A c 1 I1 cIr WEI E u xi 43 Iquot E W H 19 aw NJquot F 2 2 6 mil 1 3 rl gy quot I X r39 6 Li H E T I P3 E R p E39 2 a 1 yquotquot 1 5 0 5 W Wham a f y xfN T i5 H furm39ti04ng1 mi 2 I pU T pU 1 4 3 3 R y E M 0U y c 1 51 5 iI3ng J 39quot r y Li 3 t3yquot cy G E E 1 H19quot 63 E 4 E 39tanquot P Q I 3x E U 1939 uIJ39 quot3932 6 f y 12x 6 ml my I c ax 5in1 5 rig y 113 3 T5 f Jc 9135 1frEI p0 sin yFI1 2 3 6 r xrimfl rn5 Aa parabola P J AW p 61 mi 3 y oJ tquot F E 3 Tb 0 F Itanquote ice T C y 1 x339 39 l g inx EU 3 x139 quot 10 11 a xy2 e c 591 ANSWERS d y xzequot x2 2x 2 Cequot e y xzcscx ccscx f y x3 cxz g xy sinx sinx xcosx c h y 3x2equot2 ce 2 i y x3 clogx 139 y xzu cequot 1 a 7 x cxz b y 3 sinx 9xquot cosx 18xquot2 sin x 18x 3 cosx cx393 c 1 xy logx cxy c 1 x2y ce b x ye cy d xfy fy2 0 axy 2cequot b 3x y2 cy logy 222 cx y tanx secx 8 x1O tWl0 t4OStS10 a 45 pounds b after 13 3 E 169 minutes kxxo k2 quot kl a Ifkz k y equotquot e quot239 and if k2 k y kxotequotquot39 b About 66 days SECTION 11 p65 1 3 4 a yz cx C2 393 X2 Y 39 C22 0 c y cequotquot c2e quot 1 d y x2 cx C log x c c2 2 e 2Vcy jcx c2 f y c2e 39 g y x2 c logx C2 a y 1or3y x33 b 2 3 8ye3quot c y log2e 1 a y log cos x c C2 b y log cequot e c2 T 2JcE75 E 89 minutes 5 s so cos Vg4a t period 4JrVag I 592 J iFE HEHlI IAL EmauTnHs s EC1imIwm 12 0 71 2 pambn a y CI E m quotil39hEl39E lh lmttln m Hf the curtain Y 1en tine 1axis Amz5J the law 51 pnin1t of the card is on that ywaxisk quot A h IimnIal strasigh l iiine 0139 a atE nar3u F quot 1 9 W D a 21kE 11kc 12 S HIE distancic the ralbit runs is rkJ39I 1 1 2 E 39 1 P 2 i 391 1 M a Ely E EDEE andf the do can get E105Eff than uf2 E fair axnly F V II but rmt as close as all a E I Iquot 3939 E y i 5quot 1 quotquot If g 22 b p 3 1 than 3 a m as x as D and 4 Baal Vwill never Iandi 1 2 bfk 1 than 39 m39 as J 3 J and the boast wil l land at j f2u 4 E vs 0 I2 if Jl than 1 p as 1 gtA U armd the boat will llarmvd at the arigiirm i 0 b ms 4 m I where Lam E c L uznf a Q 3911 339 Ab case 1 P 1 Q EnCr 39 03355 Ea 33 RC P 1 id39fRE EAr w g 3 fir E1205 D E v Q E QcAosw I V w4v R sinrx n VEIILMnun E3 0Y LwiA E HE39S 3 3 3 CHAPTER p 715 Jry I Eng 3 E ANSWERS 593 3 3tanquoty 1 x 1 logy 1 x 12 c 4 yx2 y2 xzlog y vxz ya y2 3x2logx cxz 5 3y 2x2 cx2y3 6 Z yz logi c 397 y2 czez c 8 xy xsinx cosx c 9 y xlogy ex 10 ye xzy c 11 ctanquotcx y c2 12 y x2 cx 13 y xsinx Zcosx 2xquotsinx cx 39 14 3x 2y log 3x 2y2 x c 15 xcos x y c 1 16 y il0gx2 c logx C2 17 yequot sinx c 18 x yl0gx y c y 19 y xequot 2 ce 2 20 xzyz 2x3y x c 21 y x 1 x2quot39 c1 x2quot 22 e siny cosxy c 23 y clogx V1 x2 c2 24 2xe x2y2 2x2y c 25 2xequotequot y2 c 26 y x l0gx cx 27 3y cos3x 3sinx sin c 28 y x39cx2 1cxz 1 29 1 equotquot 2 cy 30 Sy 42 45y 45x 2 5x 22 c 31 x3 log y c 5x x3 33 x cx y2 32 yz log 3cosy c 1 34 logx c 1 I 35 y x2 E1l0gx2 c c2 2 2 36 x3y xy3 c 37 4x2y x2 1 cx2 1 DEFFEREHTL L ED UATIUllHE 39 b 1 1 3 E quotlngcz xyI yf ca y E I tan big car 45 cry E cmx l g clx E 1 W A y 4 tWTe 49 50 51 8 my 2 4y 1r 1 3amp 111 a xe E 11 5 p6 205 y 2 1 E I e mg J rmsyLlg 51 15 mg y s C we E a y Mex vwL I ESE Ming sen y r When I 2 7 EE r5 E 0 I y 3 mpmqz4 3 A II 3 whEne Z Ifl3939 EquotHIquot391h39339quotquotI I Iquot ii BILII m wfIn cily 1 Mag 1 g m l H H hm bI39I 39lDU39l migfht lug m E m2 a If Em c nstant 3rcJeIera39tirn dug in the mn5tVan1 g39r39auitatiDnai eld i5 f39I tn39DtEILi by A than H 1 E e39 39 r E m Ev1 I rj 39 Ea Ara W4 E y E 3JMy lib AM 4w y M 35 u2xz39 E 1 y 2 C 6312 I cEIIrg 1 E E 5323 19 13 p fl lfflxl E13tg iii ANSWERS 595 1 c y 5 sinx 5aycxc2e bycc2equot 2x 1 c y ce cze Esinx d y cx cze e y c c2e 2 2e 6 a xzyquot Zxy 2y 0 e 1 x cotxyquot yxy y 0 b yquot kzy 0 f y 2y y 0 0 y kzy 0 g y 2y 3y 0 d y 2y 0 h xzy xy y 0 SECTION 15 p91 2 y x 2x 3 y 3e 2equot 5 y x2 y2 x y 3x2 2x 39 6 a y 6e 2e 2quot c y 4e Zquot 3e 3 b y 0 d y e392 e quot 7 a y a constant or y log x c c2 1 11 a u e39 v Q 2P EP2v 0 b y cx c2equot392 2 SECTION 16 p94 2 a Y2 icosx y csinx czcosx X 1 b y Eaquot y cequot cze A 1 3 y2 2x392 y C czx 2 12 2 4y2 Zx ycx c2x 2 1 x 5 10 T 1 y Clx C2 2 g 1 x y cx39 2 sinx c2x 2 cosx a y cx C26 y cx c2xquot2 c y cx czxex 8 y cx c2xfx 2e quotquotquot quotdx 9 y cequot czxze py mFFERKEHTIAL EuwT1UN5 10 E1 3 8 y W 1 E J39 In39E x ce 39 2aJIA 1 V me cgfxi 392 c e c2a393 31 1 11 y qequotquot c2e J eI 1If4 ax SECTION 17 pr 97 1 efa 9 bill JP Q y id 1 4 y Q P E F hi 5 Iii J9 U y Va 1 k 3 H J m P in J 0 J P Jr E1 J fl J I v H y Eb 6 0 J fit It Ed y a J n J E y hy 0 y 3 2 33 quot39139 me m 3 ms E sin z equot rq crzus 0 C2 sin 0 Angel c1 xEEquot 635quot age 1 1 F e quot c ms C1 am 3 ElE3A2 hcix agm 3 29 39 quotquot e3 c5 ms 41 3 sin vu cquot1e 5quotquot c2 239539 H e r1c cm c2 sin 39I mE lI c392e39139 squot 1 ms i cgsinlxix 2 2 1313quot 239 i c1equot 1 Equot ac39E3z E quot Ehl T39 1sn c33inx me c2e 5 e quotquot l39 E E 255 quot1 7IQ1r Zsin f 2fiu 2Eu 2 Am 39 id 5 E J ry EVE E iI 13139 5xH3 x quot a39J EDS lngxl 3 sin lDgx3 at E E39Er39Wugr rx six 394 atr3 31 clt1r SEX l gx 117 393xquot391quotquot 2 IE2 I rquot u L305 lngx 3 5m s l0gxJ s E51 2 C3 1 ix 5 six 394 Equot quotquotquotl 05 viii E Sm P9x D 8 b not pnssiblxe ANSWERS 597 SECTION 13 p 103 1 1 a y ce2quot c2equot5 3 equotquot b y csin2x czcoslx sinx c y ce395quot c2xe 5 7x2e 5 dy ccos2x czsinzx 2 4x 5x2 e y ce3quot c2equot2quot 4xequot2quot f y cte czez 2sin2x 3cosZx g y csinx czcosx xsinx h y c czez 2x 3x2 i y cte czxe 3x2equot 1 j y e c cosx c2 sinx axe cosx k y c cze 2x5 10x 40x3 12Ox2 242x 39 b 2 y c sin kx czcoskx I S l I fiiunless b k in which case kx y csinkx czcoskx 1 3 a y csin2x c2cos2x xsin2x Zcosx 1 x 2x2 1 1 b y csin3x c2cos3x xcos3x 2sinx 2e 1 3x3 Zx SECTION 19 p106 1 yp 2x 4 x 2 y e 1 3 a y Z cosylx log sec2x tan 2x 1 x 3 1 x b y Exze logx Zxze c yp equot 8x2 4x 1 1 1 d y ixe s1n2x are quot cos2x logcos2x quotLt 1 e 7163 f y e log 1 equot e cl log 1 e 4 a y x sinx cosx log cosx b yp cosx log cscx cotx 392 598 DlFF E39RENTL L E UATslDH5 1 s 1 2 gap Ecnssx mgs ssscx tans H Esm log scscss tI39 1 cl yp 3rx2s1nr J cuss smxjs II 13 yp weisnrsx lag sent Ian 1 f yp J2 msx sins sint391srg mass 1 yp sinr ngcscx Entsr v cuss 1Dgsea2J tan I E fFx I ff sin I dr 6 P y suzrpr sfxE P Ex xl I3 rs quot s2x 39 1 1 1 C 3 Ed 1 s ss39 gsquot 2 1 cue CgX 1 squot r 1 6quot T dx swhers thli 139II39lTEgI39EI I5 nut an Jr 0 y r715 sax 139s39 x I xJ Elsmcntsryr fL1nctism d 2fl 113 7 V 1 r 1393quot x A 1 Thus frequency 1s a whsm 2 E xi is positive which is more p 1 rssstrijctive than this mnditi n that V M 4M3 pka 2 s 299 ssrsJnsds 4 Abmat SM pmmds 5 The rntmnd trip itims is R ssmnds whsrs B is the radius of the sarth this is appr sissmstcly 90 minutes This greatest speed is sppr Usima tsy 39 4L miiBs minsut s UI 443iL m ilssAhmir 1 1 1 32954 sm 4 39 moss 4 21 p 0 1 H P years b 39ssrss G 125 jxssrs 2 Asb ut 039 sstm139mmiss units 139 3s39IJD39U s milss U3 I39sbs u39t 295 yssrs ANSWERS 599 SECTION 22 p 127 y c cze c3eZquot y cequot e c2 cosx C3 sin x I9l 1 1 3 y cequot equot 2cz c0sE3x c3 sin Ex 1 2 1 4 y ce equot 2cc0s2 3x c3 sin23 5 y C czx c3x2e 39 6 y c czx cx2 c4x3e 7 y cequot cze C3 cosx C4 sinx 8 y ccosx czsinx c3cos2x csin2x 9 y c c2xe quot C3 c4xequot quot 0 y c czx cosax C3 cx sin ax 11 y c c2xe C3 cosx c4 sin x 12 y c c2xe e 2quot c3cosx c sin x 13 y cequot czez c3e3 14 y ce2 C2 cgx c4x2equot 15 y C c2xe2quot C3 c4xequotzquot c5e n k k3 k2 k3d2x k k3k2 3 kg 0 39 cit quot12 dtz m mm2 39 mu 2 ml 2 k k 3k 3k 18xcc0s tc2sm tc3c0s tc4sm t m m m m L Ead12 2Jr m 2 m39 19 y cx3 czxz C3 c4 sinx x4 20 y c czex c3eZ 5x 763quot 9 1 21 y 39239 x quot39 56 JC 22 a y c czx c3xquot b y c1x czxz cgxquot c y c1x c2 cos log x c3 sin logx SECTION 23 p 135 1 y ix 11Eequotquot 2 y 317928 24x 26e2quot 1 3 y 5x5e 2 2 i V11 13 19 EL L EQ U L39 HvS 1 13 61 Qxl 43 T3 7391 T M V 1x 15w1EU x 6 43 246 ms 117 9 1212 5 pN 511 l x 0 I x1 211 P Aeelwxk Eur ii 11 V EE1quotquot394r39 4 21 131 425 2E hI1 4 352 4Lr3 671 339 z 3 3 2 E14 ampx Cg Jl39 C393 39l y ax Err E y iising 231 2jl EEaf i1 SECTION 24 p Mquot 1 4Fu 39 390 413 L F ANSWERS 601 SECTION 25 p164 3 If f x 2 0 and k gt 0 then every solution of the equation yquot f x k y 0 has an in nite number of positive zeros SECTION 26 p 171 6 2nxquotquot SECTION 27 p 175 1 a y al x2 5 K aequot2 yquotquot1 D2ELQs xK 2 3 2 3 1f 1m1 x V11kT 2 a y ax no discrepancies b b y 0 y cequotquot the latter being analytic at x 0 only when c 0 132n1x2quot 3 39 Sm x x 24Zn 2n1 5 3 x yquotm39mZ39 x2 x3 x4 1 xi3973 7x le39x l SECTION 28 p 132 1 1 1 lya1x23x4 x x D ax a1 x tan x ax 2 4 6 x x 2 2 a yx1 2 4 216 X3 X5 x7 y2xxquot quot 39 3 asfysv n 1a a n 1n 2 X3 X4 x5 agtWgt123 234 2 f3T2 7 quot 3 339 an2 x my 9xquot3222lty4quot23 L5 39 ANSWERS 603 0393 I X b 1x 21352n1 n0 x y2x x 12 L x 1 Zex2 7 21 C yx xquot21 Ex Exz ZC y2x13x2x2 1 1 d yx x1 31 1 56112 39 39 39 y2x x39quot21 x x2 6 b y2x xequotquot SECTION 30 p 198 1y x214x 4x2 2 y cpcme czxme log x x2 x4 3 8 y1 39xquotsinx A 1 1 1 byquot 21 x 6x2 g6x339 1 1 1 y2x 1 xixz x 339 5 X4 8 2 y21 cosx x2 x quotYquot1 2 2 272I2mquotquot3939 2 4 Syxquot21 m xquot 2sinx 2 4 n2 f c 12 y x 1 24 x cosx ANSWERS 605 sin3x sin5x 3 5 sin3x sin5x 3 5 sin3x sin5x 3 5 4 6 a 1rEsinx b sinx c sinx sin 3x sin 51 H 3 5 sin 32 sin Sx H 3 5 7 After forming the suggested series continue by subtracting from the series in Problem 1 then dividing by Jr 1 6 d 2 smx 3 2 e gtsmx SECTION 34 p263 2fx 2i 1i5i n jf39 It 1 221 quotquotquot 12 Jr 2 39 cos Zn 1x sin Zn 1x sin Znx 3 3 K 4 M2 Qn D2 2 m 1 2 h 1 12 In each case 2 Zt T5 sinhiyr 1quot 1quotn 5 fx It 1 2quot 1c0snx 22912 1smnx SECTION 35 p269 1 Even odd neither odd even even neither odd 1 1 Ir 4 1 3 3 I this concrete sum familiar to us from elementary calculus provides strong emphasis for the very remarkable nature of the sine series we are considering as x varies continuously between 0 and Jr each term of the series changes in value but these changes are so delicately interrelated that the sum of all these variable quantities is constantly equal to astounding U 2 4 ncosnx 5fgt Zlt 1gt4n2139 2 4 2 6sinx Z 0SxSn 4 cos Zn 1 1f E n U2 E UATiiDH395 Q Q 13 31 1 E v 1quot V I n p quot T V P VT A En 115quot 10 b4 I P a1 39 E E Ix pO Jr J W E W 1 1 A g 1 F 2 39 215 3 1 T 121 in 12 1 Q 4 u G 1 1 PE w g 3 Fl 1Ecu5 2n T 1 bl M I J1quot p a in 1ms2n 1 my a 1 g ms x pw ms 2 2n 1 ms 39 E 1 T SECTION 33 p291 2 p w 0 g P Z 0 P Ir J y39amp39 b X 2 b 1 bi j P L31 Pe Pe Pe 1 fa A 4n1yx39 i sin 2 tr VA sin a U5 191 sin Zn 1r El 5 1 0 E 1 Z 3 sin A 1 1 vi 1 E IL ANSWERS 607 c Aquot nznrz yx sin mtx nznrz mrx d 1 i2 yx smIJ nzirz mtx L C A I 4L2 yPx Sln 3 n2t2 nrrx a 0 An39393939ba2ynxS1n ba 39 c n1sin2n 1 cos Zn 1at 539 a yxt E2 1 Zn 12 7 1x cos 2n 1at 8 39 Zn In yxt quot Zn 1 3 c yx t G 212 sin 31 sin D sin nx cos nat SECTION 41 p316 2 wxt bequot392quot239sinnx gx where gx w 1wz w1x and b frfx gx sinnxdx It Jr 0 2 4 wxt equot S bequotquot2quot2 sin nx where b G fx sinnxdx 77 0 8w QB 0 E x axlgn 0 wxt 100 2 2 quot cos nx where 1 6 wxt 39239a E ae 1 2 quot aJffxcosnxdx forn012 0 on 39 2 7 39 7 wxy 2 be quot s1n nx where b 0 fxs1nnxdx 1 I 0 SECTION 42 p 322 hrquot cosn6 4n2 1 2 a wr6 2 0R 1 1 1 b wr6 2rsin6 Elr2sin26 r3sin36 K D4mEREwaL EQUATIONS quot 9 I 1r1 39 1 grain 8 Ella H J J mgr ha d wvfr 9 5 hi rain 9 r3 sin p K r sin SH I H y r as n E WEN pi 0 NET hi p 43 p In H 1 1 y L391quot jJ 3 1 IW Em mp 1ju a E ratquot 1 0i 1 I3 FEM Cl 1 4 ram p PEA 1 0 W M Vlw E Emu d Us 6 1394 M4 U K xuquot p 1y 1 PMJ 5 U V I b Lg nd r quot ss and Airfa 3 3 1 39 1 2quot39139r 4 MP 11quotquot A9 d E 39yf 2p Jj lift W Iv 0 fin My pa 0 II b 113 Ch ygr 2 1 1 2 fIl fm ns 14 0p and the eigcnfunctiuns cnrr esp Ending Vto each Bf these Z are cums 1 and sin 3 PM 3x1 a 1 F x SE 3x 0k 41 1 2 3515 E a Pg x E w3xL5 V r3 A 3915xi ANSWERS 609 SECTION 45 p 347 1 1 5 4 3 21Px Px 15P2x 39 39 39 b fx e equotP0x 3equotPx Se 35equot39P2x SECTION 46 p 356 7 y x cJaxquot c2Jax ifp is not an integer y xquotcmJax c2 axquot in all cases SECTION 47 p 363 3 J2x Jx J0x J3x 83 quot39 1Jlx quot J0x3 mx 3 Jltxgt x 1Jltx SECTION 48 p384 4 1 1 p 1 1 p 3 Lsm2ax 39239 d P2 4a2 and LC0S2 ax 39 quot the sum of these transforms is the transform of 1 1p 10 4 2 4 8 d 4 p p2 4 p 1 5 p 6 b f p6p24 ep7 2 5 cp 3p225 5 a 5x3 d 1 equot b 2e 3quot e x sin x c 2x2 3sin2x SECTION 49 p388 1 equot 1 p 2 39 quotquot quot a 126quot C P20 1 1 1 equotquot b pequot 1 d m 0 0 E 39UJL17l39DNSi 5 j p 3 arr 21 F L 1 9 A P1 P 113 alt p sin 3 c 3 05 m 3 sin g g 2 quot1quot39x1 2 fa mm e 2 Lb p p 3e as y x 1 e39 r0s4x d y xf S E 311 4 71 523quot l e yJ 3 am 339quot simz 4 p Mquot N5 ame 5 1 E 5quot 4 1 as 1 Aefl slmz if 3 L y 2 M m a O C05 11 a T 1 1 sir1 0 L L Lip HT 2a1 ip 2a3 r 1 4 39 3 391f cxE e bi U 155 L b N b S 3 lag Jib tanquot 5 ID pi quot quotI39 3 yx 2 ms 4 C M r equot i39r 1jlIE b N3 E1 1 yfxj 2539iElI1I 4sinlr y r M 53 p411 131 max 1 E 1 a at 11 Am p 5 E b2 3 5 E n E El r a sir be b sin HI m1 ntEwnL Equot5UA lT N5 8 8 8 P ce Acie T 7 gr 55 T i gr cyH q 3 32 2e 1quot 1 E i E 1 3216quotquot E3equotF y 3cIe quot r1 3rquot1 Aquot 27Tc1Equotquot c2 equot St E 2 y 3cKle39quot 39 E355 12139 3 5 Th E3 I J5 E19 af2re Q n 5 M I aj SECTION N Pi 4133 L H r 2c1 quot a39quot139E EIE L cgge 4 e339V2c 5 sin equot c cn5 3 3 sin 3 c45in 33 e 3 cm 3 T 1 2equot39 4 Egil Ze139 1 Elf c 2r 139A4 H II M quotH1 3 3 J K 1 I C1932 Eur V dam 2 393Equot uh ad H2quot 33939 w i E T EEEur E 391 ll i H II E rmf139V cite 1 I1 H 3 J E V2t39E39 BCEAEE I 7 7 cm 39 a 2523 1 w H Ii M 393 E c 05 E 15iilquotI 2 e391 cEsin Ams As cgisin cg524 G V 3 5 b J J y w 2 391 an W hr 1 E 2 2 2 3 A ix M P s p O V ix 1 2 ampf2 x 1r k as air r 2 The Em curve is r nrwamp up whenever the rabbit curvve is risiirag ANSWERS 613 SECTION 53 p 445 2 Put c t 2 and use uniqueness 3 They are the same except that the directions of all paths are reversed in passing from one to the other 4 a Every point is a critical point and there are no paths b Every point on the yaxis is a critical point and the paths are horizontal halflines directed out to the left and right from the yaxis c There are no critical points and the paths are straight lines with slope 2 directed up to the right d The point 00 is the only critical point and the paths are halflines of all possible slopes directed in toward the origin 5 For equations 1 and 2 they are 00 lJr0 l2r0 d3r0 and for equation 3 00 is the only critical point 639 3 10 020 10 222 7 x ce39 yce e c2 SECTION 59 p454 1 a i The critical points are the points on the x axis ii dydx Zryxz 1 iii y cx2 1 b i 00 ii dydx xy iii x2 y2 C2 c i There are no critical points ii dydx cosx iii y sinx c d i The critical points are the points on the yaxis ii dydx 2xy2 iii y 1x2 c and y 0 2 a 1 iii xy c ii dydx yx iv unstable ta 0 252 in y ext ii dydx 2yx iv asymptotically stable x 2c cos 2t 202 sin 2t C 1 y c sin 2t czcos 2t dy x dx 4y E14 DIFFEREHHAL E UAquoti39iD lW5 xi y V 1 iii 4431 c Iis39 stahIs but mat asjrmp tustisally stable ECTlDN 60 2 1 a Uastabia ands b Lsrmptnti a1lty stable s39pi139als Unstable szaddls paint d S tahle but must asympltnticam stable centerquot as Aasymptaticaslly stable mas f The critical point is not 391sslssteda g Unssatablc spiral Pj 5 The Irist ical pnignt Qa3gt2 than transfusmsd system is and the csritical paint is an asyampstrnatsisally stable u r Z mi Ebm H3 H p 39s 2b 4 12 13 I0 stable but nut aasymptatisssa y stabltc sezntesr the mass ascillatss tihs uiisp1assmant ac and wallaciaty sir are psrrisrdiis fu39nstians sf time ii An asymaptntissally ssrtabrle spinam the mass executes damped ssillav tisans s and saxm U thrmsgh 5 m Mf39a1 and smallalsr DssiHatim1s iiiiI An asymptnticalIy stable node the mass dues not nscillate J and said s H atithnuat nsscilsllating iv 39I he same as iiiiu k sax E mxy 1 bi c SECTMN 61 am 1 a Nsithsrg c Nsithsr I3 Puasaistiva da nits rd Negative czls nise SECTIN 62 p 47939 2 s2x ya dx i 2 lay 3 Put D pqr 11 ib2 1b392 azbl Us 4 Na sainalusi n can be drawn Kabaat the stabilistjf priapsrsias of the a0nHnsar s391equotsIIsn39m 4 at 00 when atha 39r s1ata linear srs tsm 3 has a saatar at 5 a Unstable spiraE CD Asymapst tically ssstsabla nna The critiaaE p 39i139quotlt 0 is unstable if gm 1 and sasymjgtptUtisaIIy stable if p 2 D ANSWERS 615 SECTION 63 p485 1 If f 0 O and xfx lt 0 for x 0 the critical point is an unstable saddle point 3 y x2 x 2E i20 is a center 00 is a saddle point and 520 is a center 4 When 2 F x has a maximum the critical point is a saddle point when it has a minimum the critical point is a center and when it has a point of in ection the critical point is a cusp SECTION 64 p492 fir r4 r2 2 a 4 quot 4 dt x 2cos4t to x 2cos4t C T y 2 sin 4 2 sin 4t to ky 1373 4 a A periodic solution Li nard s theorem b No periodic solution Theorem B c No periodic solution Theorem A d No periodic solution Theorem B e A periodic solution Li nard s theorem SECTION 66 p513 1 3 x C2 Y2 Ci b y c sin x C2 2 y x2 x 4 a c r cos 6 C2 b Same as a SECTION 67 p 523 32L y2 t a2b2c2 a2b2c2 cd M 39 5 The catenary y A c1 cosh x E C2 1 ii 11 391II3939L E39 C1 ymx A 1 x1 ygIC39 1 J 22 Ex y3iI 1 I 1393 I3 5x 3155 3325 h p E33 1 III 392 ym p Q HI 2 MI g E PS 2 11 E x 0 7 V W A J I4 I InFI 9 1 J Er quotquot J 1 M E y1I 3iI1139 I 1 I 0Q V A 1 A1 L P T 3 y3I r0sr 1 1 17 sx393 4V 2 3 H3 P V I y339 s1nr x 1 I xi V xi xiii A Mv P T xii Zquota 394 EDSI 1 B 1 I p H SEACTEON p 5552 1 0 b by All Vp aim5 rr y sim39e Vfry 5 y Lsatis e 5 a 39Lipschitz cmmitin an Everjr IE Et g AIM pnims 1ETg jr39AgA SECTION 70 Z u 1 y uc5n E 2 nm II INDEX Abel Niels H 89 221 223 230 336 413 495 formula 89 integral equation 403 mechanical problem 401 quoted on Gauss 223 Absolute convergence for improper integral 385 Achieser N I 235 Action 527 principle of least 528 Adams John Couch 183 Addition formula for Bessel functions 378 Adjoint equation 329 Admissible function 505 Airy Sir George B 183 equation 183 329 functions 183 Amplitude 107 Analytic function 171 Andrews G B 142 Andronow A A 481 Arago F 137 Asymptotically stable critical point 454 Autonomous system 442 Auxiliary equation 95 123 428 of a system 456 Auxiliary polynomial 123 Ayoub Raymond 276 Ball W W Rouse 149 Barrow Isaac 147 617 Bell E T 140 227 Bendixson Ivar Otto 489 Bentley Richard 150 Bernoulli Daniel 42 299 306 Bernoulli James 42 138 Bernoulli John 36 42 43 137 151 502 Bernoulli equa on62 numbers 275 polynomials 275 solution of wave equation 307 Bertrand s postulate 237 Bessel Friedrich W 228 229 348 380 Bessel equation 3 161 163 164 198 199311 329 348 general solution 353355 generalization 357 normal form 161 163 of order p 185 of order 0 191 395 p 0 solution 172 191 350 396 second solution 199 p 12 solution 94 198 357 p 1 solution 198 350 point at in nity 207 vibrating chain 309310 vibrating membrane 371377 Bessel expansion theorem 361 Bessel functions 348 358 addition formula for 378 rst kind of order p 350 generating function for 377 integral formula for 379 integrals of 360 Convergent series 167 Convolution 400 Convolution theorem 400 Cooling Newton s law of 28 Coordinates generalized 530 Copernicus N 150 Cosine series Fourier 268 Cotangent Euler s partial fractions expansion of 271 Coupled harmonic oscillators 125 Courant R 145 284 356 535 Crelle August L 414 415 Critical point 443 asymptotically stable 454 center 449 focus 451 isolated 443 node 448 path approaches 447 path enters 447 saddle point 449 simple 472 spiral 451 stable 453 unstable 454 vortex 449 Critically damped vibration 110 Curve stationary 508 Curvature mean 535 Curves integral 8 oneparameter family of 8 pursuit 66 Cycloid 40 4344 65 404 512 d Alembert Jean le Rond 299 302 formula 309 solution of wave equation 309 Damped vibration 108 Damping force 481 Damping linear 480 Darwin Sir G H 495 Dating radiocarbon 2224 Davis Philip J 301 Day W D 409 Decay exponential 21 radioactive 20 21 Dedekind Richard 225 Degrees of freedom 530 Delta function Dirac 389 Descartes Ren 41 Differential equation 1 complete 84 exact 51 linear 60 normal form 159 INDEX 619 order 3 ordinary 3 ordinary point 176 partial 3 reduced 84 singular point 176 184 irregular 185 regular 185 standard form 159 see also Equation Differential exact 51 Diophantus 140 Dirac P A M 389 delta function 389 Dirichlet P G L 240 256 260 262 301 414 conditions 260 kernel 293 problem 228 317 370 for a circle 318 theorem 260 Discontinuity jump 258 307 simple 258 Discretization error local 563 total 564 Distance 281 between two functions 283 mean 121 Doubling time 20 Douglas J 535 Dunnington G Waldo 225 Dynamical problems variable mass 7880 Dynamical system conservative 480 e 17 Eccentricity of orbit physical meaning of 119 Eddington Sir Arthur 420 Eigenfunction 219 303 324 326 Eigenfunction expansion 307 326 Eigenvalue 219 302 324 Einstein A 226 243 284 370 441 496 503 on doubting the obvious 370 on future of mathematical physics 441 and Poincare 496 relativity impossible without Gauss 226 on Riemannian geometry 243 special theory of relativity 79 use of calculus of variations 503 variable mass and E Mcz 7980 Electrostatic dipole potential 370 Electrostatic potential 366 D max E397Hip5vE 11 T19 Ellip i iintEgEaI rstg I nd 5D nd kind 32 EnErgr crnsenratiun Ant 3111 531 kincticg A IeI1liaL 535 EuaEinnjs Abgl s integraI 403 Airy 5T IE3 329 auiiiam 95 123 423 Bf 39EI15jr5tEm 456 EEiT39i139D li JIiquot5 62 es sel s snag HE sE139s cqiuatriikunj IChehy s1l391 ev39s 391 33931 203 329 L334 cumpi339t e 4 diEeIn39tial see Di emntial quatinn zquidimensaiu nal EulEr 5 983 I33 161 136 319 EEsullerquots Eur IE l U1L15 f 39eraI1ia39tim15 S Exact 51 hea39L 3 311 Harmi te 5 M3 2111 32399 a hamcrger1E us l8 h3rpequotrganamet c mn uenru aussfsi Ph gsEner39aiiEed MD indii a1 p 195 tiznt yal SEE Integral snq39ua39lian Lagrangie 1s 531 Lagruerres 28 0 334 LapAlas s 3 16 3amp5 Leampgenire391 3 LITE 325 3529 LiEnam39s 491 39lin2Ear Eii lamen I ziaL uf mDi l3nf1 fur undampcd pencJ u1urn EDLEEE 4732 nu nhamwgenE us S3 nndimEn5i nal ham I313 minedimts siunal wave 293 304 P arsmnail 239 24 prnpredatnr VulLerTraquot s 436 39r educe1 k Rima39Ii 536 RicLcati ueq uati nA 39Rjtjmampnn 5 Schr ii dinger w aan 219 second nrde1r linear 31 sewa djAoint p 5aparahiEi 5 SturmLiuvEiEe 333 aw tr dimensinnal Laplace 3139Zr39 van d r Fail 441 145 492 wave see Wave equaitinn Eq Mirn nsi naI ecwa1tiun E uIerquot5 98 1213 MI 1 319 Equ ibrium epoint 443 Eqm1iibrium papulamiuns 43 Em lyi A 166 EEID o Errm hirr 11lar f p nd uZlum clmk5 31 lracal discratizatiom 1 5673 tuutal li5crediatiurn 564 tmaa relatiw 561 Escape velncit3r 134 EuscIid s thswam 14 Enigma hsennhard 39136a146 202 3 243 252 quot236 2939 L 323 336 SM 5215 526 characteri395li 145 circuit Al212 constant 1391 aquatinn fr EalquotIJl L1S uyf var739iatin39n5 503 eq7uidimensinna1 equatiun 93 123 161 319 f I mLIla5 for mmp ei numbar5 fnr Fauna rtnm cients 243 301 P f 1r39puulr39hedra M4 hjrjpcrgenmaalriCEuI1Gti11 EH2 idEn tr Eur pmi3rn5 Pq iF1 1ni39tn pmdmit fur the 5im 2 irratiinnality M E 323 and iLagrangei law of quadratic Ieciprnui ty 213 mttlrmd 56 m th imprnwad k rnimmal surfamzgs 5334 partiaI framinns e xjpan5iIan mquot the ta39ngent 2 path M2 on stquusncc DE prim a 235 srumg emf seria5 133 3363 O rIhenuem an humugen emus funu39tirms 53quot vibrating me mbrane 313 Ewing 0 0 5135 LExm t d1i em1tia 5 Exam equati n 51 Expansiian ignfunli n 3UT 326 E pan5i nn HB quot ifidE 32 E 3pamsinrm menr e m E tE539EL 35 Heavi395id 411 La39gendrE 345 E p nentEa1 dca3r21 gruw1hj 19 mi l3Eur fur1utinn DE 33 shift ruE 113394 E394pm1En 195 Extremal 503 FVaims J D 563 Fall free 29 retarded 30 Fermat Pierre de 41 principle of least time 38 536 Fermi Enrico 80 First order reaction 20 Fischer E 291 Focal property of parabolas 59 Focus 451 Fomin S V 505 549 Force central 116 conservative 526 damping 481 gravitational 117 restoring 481 Forced vibration 111 Ford Henry 114 Fourier J B J 255 301 311 coe icients 249 279 307 series 249 279 301 cosine 268 sine 268 307 FourierBessel series 361 Fredholm 1 328 435 Free fall 29 Free vibration 111 Freedom degrees of 530 Frequency 108 natural 111 normal 127 resonance 113 Frobenius F G 188 method of 188 series 188 Functions admissible 505 Airy 183 algebraic 165 analytic 171 Bessel see Bessel functions bounded 258 Dirac delta 389 distance between two 283 elementary 165 even 265 exponential order 387 gamma 351 generating for Bessel functions 377 of Legendre polynomials 341 harmonic 318 Hermite of order n 216 homogeneous 48 Euler s theorem on 531 hypergeometric 200 con uent 207 inner product of two 282 INDEX 621 input 405 Legendre 180 Liapunov 467 negative de nite 466 negative semide nite 466 norm of 282 normalized 277 323 null 282 odd 265 orthogonal 323 345 sequence of 277 output 405 periodic 250 piecewise continuous 386 piecewise smooth 296 positive de nite 466 positive semide nite 466 Riemann s zeta 244 Schr dinger wave 291 spherical Bessel 359 stationary 508 transcendental 165166 unit impulse 389 unit step 388 Fundamental lemma calculus of variations 508 Fundamental theorem of calculus 6 2 839 Galileo 36 43 150 Gamma function 351 Gauss Carl F 136 202 221230 236 240 241 242 262 322 348 366 413 414 526 536 537 and Abel 414 complex numbers and quaternions 536537 hypergeometric equation 199 hypergeometric function 202 potential theory 366 prime number theorem 236 Riemann s dissertation 241 Riemannian geometry 242243 Gauss Helen W 221 GayLussac Joseph L 413 Gelfand l M 505 Gelfond A 0 328 329 General solution 8 84 88 Generalized coordinates 530 Generalized hypergeometric equation 240 Generating function Bessel functions 377 Hermite polynomials 214 Legendre polynomials 341 Genus 145 Geodesics 510 on cone 514 GndesisacanmT an cylindscr 15 L in iphyrsim N an spTl1m 514 5423 G ijbb5 J Py Elba pmpamties1vnf paths 436 G uldst4ine HEEVMEH 551 Gradjiant a Graph 14 G1afph thnr3r H31 Grassmann 53 GrarifIm in n KNewtnn 5 Alaw uf 117 419 Gra vita m1aJV zrsnknstant ll f 39c 11139 pmrenmiiial 366 Gar Aw 339iquotf 33 39Gren G1nrge D Gmwmi P e13pmJnn39t7ia1 3919 p fpl1li I1i W Hadamard L 72435 Hald anE JL P 5 Hailfeme 21 Halle5r Edmund 1HE Hapcrim L 339 Hamiltcrni Rnwam 536 HE1391391i ED J395 printipI 243 SUE 52 Hiamming p W 553 Hard3r Hg 5 14232 Harmnni E1J Eti ME Hjarmanie nsr latar IND 213 muupled Harmn39ni vibra minns simp4Ie 1amp1 HERE equatim1 3 311 andimnsi rn am 2 l eat sp i w O Hs1avisid ej livmr 123 ti axpan5inm EKpa 5iEI theurem 411 Hegeal h WW HEr39miteV Charles pf 323 415 ezqiuatislm I33 211 329 334 hln linns of n1dexr mg 216 nvrth gunalim 25164213 palmnmiaIls 9 211 pr gmcr39a39Eing Euni39EiDn Bf E14 R driguE5 Iinrmuia fan E716 SE139iE55 HE H39q r5rhul Sir W lJiam 15 Hrsh Rauban am AHeun Karl 0 Elum s mvstfhmrh 5 z m z p Mz Hubba 39TI39huma5i 151 H39nmugnenus ndiIErmns 326 asquatimn 43 L33 P P P Euler39s t39haemrm 531 Iintar sy5tm 1 IImsukg Ruben Z1743 Humtm1dt HQ 0 vn n 113 Huremim 433 4795 Hurlem J 0 H39uyg n Ghri5tiaan 43 f grpzerbula 111quot quatiwm con ugnti T P P p gn raJied y H3vpeVgcommric mctaimni P m n ucnt T Hjrp rgeaumetrir series 200 I den rnr39 Eualr395 Em parvimgs 2319 I mpmp r i tEEIaJi ab5oluI unvergenm 335 mmp1a1 5an mi 386 can wrgnnrn 333 335 Iimpmwed Eulmzr mgthnd 566 t 403 Impulse fun nn unit 339 Almipulsivg respransrV Mil lndexi493 Ineditzial e wMinr 139 195 T 4 P P isaprimt SEU MLinkuwski S uhiram s Isn n39ity at 205 Initial mmdirtian B i Imiiltial value gprah mem5 33 333 lunar pm dlucIt nf hm fumctinn5 282 fInnnr product DE two vecfnurs ZED Inaput 39 JnEtinm dwiil Intcgral tuning 3 In gra1 elipticz rs139kind 3392 swzmrnd kind 32 IngraIuqfuati1rm 1IIM P PnLhEl5 4UC3 Integral ifmmuIa E55e395 3 Integral improper convergence of 383 385 Integral Poisson s 321 322 Integral transformation 382 Integrating factor 55 Interest continuously compounded 18 Interval closed 82 of convergence 169 open 82 Inverse Laplace transform 392 Inverse Laplace transformation 392 Inverse operator 130 Irregular singular point 185 Isolated critical point 443 Isoperimetric inequality 520 Isoperimetric problem 515 Jacobi C G J 230 240 336 495 on Abel 415 495 and Gauss 230 Jaeger J C 135 Jeans Sir James 420 Jump discontinuities 258 307 Kac Mark 410 Kant Immanuel 150 229 Kellogg O D 370 Kepler Johannes 117 150 Kepler39s law rst 119 second 117 third 12 Kernel Dirichlet 293 of integral transformation 382 Kinetic energy 30 527 Kirchhoff Gustav R 73 Kirchhoff s law 73 Klein F 224 Kolmogorov A N 549 Konigsberg bridge problem 142 Kruskal M D 557 Kummer Ernst 262 Kutta M W 569 Lagrange Joseph L 412 524 equations 531 multiplier 517 523 variation of parameters 105 Lagrangian 527 Laguerre Edmond 207 polynomials 208 equation 207 329 334 INDEX 623 Lambert Johann H 27 328 continued fraction for tangent 336 380 law of absorption 27 Lanczos C 226 Laplace Pierre S 412 536 equation 3 316 365 twodimensional 317 transform 383 inverse 392 transformation 382 inverse 392 Law of absorption Lambert s 27 conservation of energy 30 531 of gravitation Newton s 34 117 412 419 Kepler39s rst 119 second 117 third 121 of mass action 26 of motion Newton39s second 1 116 527 Ohm s 72 parallelogram 285 of refraction Snell s 37 Lawyers 223 Least action principle of 528 Least squares approximation 346 Least time Fermat s principle of 38 536 Lebedev N N 345 368 Lebesgue Henri 247 291 Legendre Adrien M 223 336 380 414 415 equation 3 178 327 329 334 expansion theorem 345 functions 180 polynomials I80 337 applications 369371 generating function 341 orthogonality 342 Rodrigues formula 340 series 344 Leibniz G W 147 148 152 rule 407 Leigh E R 435 Levinson N 327 Liapunov A M 465 475 function 467 Libby Willard 23 24 Li nard Alfred 491 492 equa on491 theorem 491 Lindemann F 328 Linear combination 85 422 Linear damping 480 Linear differential equation 60 second order 81 Linear spring 480 Iwmax Linea39 5r5I m1 hnmnganH us 421 nanhnmngeneuus 421 m71mupled 431 Linaar tranzsf rmat inn 332 Lineariati nn l mgtthud 0 433 Linrarfigr depensdcnti ET 124 LincarlyindEpendEn1 ET p o Lic suville J39n5 Eph 937 32 359 Lip5Ehi p Lip5 hitz can d39itinn 552 L ba hevsaky N 63 Luca discretia14i n n crmr 563 LL mk Em J h v ISM LagariLmmic dr remenL I L rcntz 235 Ln39tka A 1 435 LuthLeramp Ag ETE Majm cases mr cm39ial poikn 456 MVan ueI Frank E 154 Mass a1 im law mi E6 Ma Ehews p U 3 331 Maxwem Jamgs Clerk tam J 33 Mead P Cji 5 325 Man c nvgr4gnce 3816 M ane E uW E39m39E P Mcan dJi5tanm 121 MeZnanical pmmem A5lIl 5 40quot MEv2hanis ti dEtm39mimi5m 420 Membrane BM 1rib139atiu1g1 Eu1er s tt 1e rj anE 3TsET Ma h d nf Frmiban iIsIs 1m Mc thnd nf 1inea4rizaIimarl 43quot18 Mtthmii f 5egpa r39atiinrt cmE variables 314 311 313 363 EM Mimod of 5urces45ivg appmximatinns T Piard39s3 MeEriE 5paQE E33 M iIilkan Robert 1218 Mirlitmal surf39a5 EuEEr39s problem mi 534 Mjin ima1K Pf p rtjf f39Che7lta45i1e1ar prrIvs rmmEaVi5sk 234 4 M39ink w5ki H ermaTnn 234 in3quaii39ty 2332 Mix ing Z1 Manr hvead L Mmi nz equa1insrf fur umdamped pend39ulrum 33 33 132 Nawtvrur1quots 56gt nnd law tif 1 116 52 Multip iEr1 ampIagrangEa 51 ki Mu1 tgterm Tajd r mampthuud5 568 rt bu39di5r pmbl m 51 N amral freqfu nr 1111 N gaIi39ir1e rde nil I39umJEit un 0 I egatimA s1m ide niIE funrriiun 4661 Nawman 11 1453 Newmn Esaiac 41 121 l 15 ME154 M2 NEW 0 E1DIfiig 0H law 1I gravita n4n 34 1131 14 12 4119 setland law Elf mmi un1 1 1 116 52 N de 309 311 riItimJ point H nnhnAmmgennus equatAian Musnhomagemnua5 linearquot 5jg3951Em 421 fNunIrin1ear m thanir5 basic qj il ti li aft IIEl N mlinaar spri nag hard 435 snfrh Nam if a fun tim11 232 N rm 1 a 1 39c rE r3tr39A Li Nzr39maJ fgrin diEerEnliaL1 qua1i0n F159 NQImaI frequenci5 I Norma iied fun39tinn5 2 i391r739 323 0 J M11 funiiII1 X Numbers EemmlIVi N5 D mB lh 39 5lE Eula Heum 566 imprinsredI E39uier 5amp6 multit2r39m Ta5vlsmr 5amp3 prcKdticIadmanrrecInri 566 iRungE Kutta 5 9 i EiE5 l EP39 553 Ghm PHF PHF D Ohm iiawi T2 U ne dimensimt1aiA hm tzquartiun 13 Cl I1dimensinn aI w aw 3 ET31 i 39 quotdilquotI1I39T5I i LI1 l Awawe equatmn 3134 One parametir famiI339 Bf curws E Dpen in39lervaI Dp eralo r di iern39tiaIi 129 Fl139El 1E E I i 0rd ur of d ar mtiai eq39uatinrI 3 mpanemtial fumrtiun GE 3 UfvdiI lj39 diEE 39rltnn tial equaIiun 3 D1gtd n391naW Njnt H6 filirag M5 Dr1h11gnnaT Eur1Etiun5 3123 34539 stquanwe vJf 3 grtlmgma rtrajec Laries 1amp3 tUr39Ehugn n4al veuztm5 EEEI Drthngnnal ityV 255 BEE SE functimls 3ltI I 3 3 ilhcbysimev jmVmmiais 0 Hermite functions 216218 Legendre polynomials 342 Orthonormal sequence 278 323 complete 290 Oscillator harmonic 107 218 coupled harmonic 125 Output function 405 Overdamped vibration 109 Paraboia 119 Parabolas focal property of 59 Parallelogram law 285 Parameter 10 Parameters variation of 104 434 Parseval des Ch nes M 291 Parseal s equation 289 290 Partial differential equation 3 see also Heat equation Laplace s equation Wave equation Particular solution 8 Partitions theory of 141 Pascal B 41 Path 443 approaches the critical point 447 enters the critical point 475 Euler 142 global properties of 486 Pauling Linus 26 128 Peano Guiseppe S48 Peano s theorem 548 Pendulum undamped 3033 482 Pepys Samuel 150 Period 107 250486 Periodic boundary conditions 330 39 Periodic function 250 Periodic solution 486 Periods of revolution of planets 120122 Phase plane 442 portrait 445 Philosophers 224 229 Picard Emile 540 method of successive approximations 540 theorem 8 418 543 Piecewise continuous function 386 Piecewise smooth function 296 Planck Max 528 Planetary motion Bessel s studies of 348 Planets periods of revolution of 120122 Plateau J 535 Plateau s problem 535 Poincare Henri 221 440 473 489 494 Poincar Bendixson theorem 489 Point critical 443 INDEX 625 asymptotically stable 454 borderline cases for 456 center 449 focus 451 isolated 443 major cases for 456 node 448 path approaches 447 path enters 447 saddle point 449 simple 472 spiral 451 stable 453 unstable 454 vortex 449 equilibrium 443 at in nity 205 ordinary 176 singular 176 185 irregular 185 regular 185 Pointwise convergence 286 Poisson S 158 321 413 integral 321 322 Polya G 142 145 515 520 Polyhedra Euler s formula for 144 regular 143 Polynomials auxiliary 123 Bernoulli 275 Chebyshev 183 204 230 minimax property of 234 orthogonality of 233 Hermite 211 213 generating function of 214 Rodrigues formula for 216 Laguerre 208 Legendre see Legendre polynomials Population growth 19 Populations equilibrium of 438 Portrait phase 445 Positive de nite function 466 Positive semidefinite function 466 Potential 365 electrostatic 366 electrostatic dipole 370 gravitational 366 Potential energy 526 Potential theory 316 366 Power series 167 interval of convergence 169 radius of convergence 168 PredictorCorrector methods 566 Preypredator equations Volterra s 436 Prime number theorem 236 Principle of conservation of energy 30 531 C WEEK 39LEinipI le smwd Djriuth e39t 9 HamiImn s 243 503 52339 Elf least atiurL SITE gr lva5E 39IijmE FEIrma39l 5 33 5315 M 5ugpcvra5Ii IiDn W3 W3 rbITeru1 Ebl s mciIl1aLmial si li aiI pre5sur 127323 hEC Il3lquoti head on rzirElie 45 ahaundary39 v39a39u 32339 ireguIar39 32 sin g ujilar hrra4Lh39istrAEhrrne 36 42 15L P 512 b ne 2 21 2 T bugs an uablE huy 113 chain an labile 451 athmical mami un 1lepsdra E i1f nETa1I EDni 44 deslzmgram hIm39ting 5 u bmarin x DirihIem 3 1l39T r STU far 3 tiraleA 0M datg rVahh4it ESQTJIA cartlim g1pii udes T122 stsci apc 31 faII2ingA rainrlmpAi T9 f lbMill 44 T geodesics Em mntx H mm rylind7er 5315 ml 5p here M hangrng cihain 0 H 5211 heal drilled Ehar39 ugh tearth 35 M3 iniIiaJ value 33 Est periamgt ric 5amp5 Kf3n39ig5hrg bridg g M2 LambErt 5 Naw niivtf a39bsrmE39ifun 2 law of mass iamiinn 26 minima a 5urfa Ce Eu lrquot 5 33d of 39rmrnluIVin Sll h l mirrm39 5S59 maIhball A44 nah djr 419 Newm n 391s Etaquotav utf cming m1edirmcnsi nal wave 309 path at b l ma1A Fla Iauquoti5 r EAresidnt and Pr39iame Mini5Eer 46 radiaa tive dwa3rE 2011 radon se39pagag rE ativi y T is i ruckgh 3939 mp wnund mrnunx i pcmt 45 rutH39tAiang can surfquot water 45 snmaplfaw P Srt u1maLinriu rille 326 rguJIar Y 5ingular Iiianlk 44 IEapeI d t lulmn 46 ta39u t clthrDnej 403 4114 tern 1inal vtlmziity 33 TInrriMi s aw 44 TKrr39irsMiquot 5 Ehenrm 43 Emftrix h mrmgl thmugh Larth4 I113 vibrating Ehaim W9310 wreak 39Ehmrm 43 PSEud 5phEIE 63 Pure re nnanw ll 13 Pur5uit rurves 66 P t hagsJrrgtan t hw rrm Duantiz d etnergyr ElmvIs RadiaEtive dar 0 21 Radiaarbon daming 2214 Radivus of mnvrgem 163 Rad T4 535 Radun saeejpagve FainviH D MD Rapaport Ana39ml TE Rat wrnstant 21 Rali I5li 1613 R a timaani rst m dser 20 asrcnnVd urder 26 Rursian farmu 3 a IE2 EhreeeIrm 133 t w39tm1 Remtiun Snc fs aw mi 3quot FuEguKIar paIrh adIa 143 Rcgumr singuIar gpuiim 155 R EguTIar Sturn1 Limwilll pmbI m ZRIaVIi1r Maren lI E L 5611 REIE iFi i Einsmainquots sLpuviaI mhemjr Hf 0 ResuKnainre 112 frc qunc3r ill 13 pur e W 13 Rspanse im p39ulsiw 499 ind39ic39iaI Rwmnring Vfurca 43 R IEardd Ea1I Ri E E FIi J T qua uniTT ape ia1 3amp4 Riemann Bernhard 136 240245 247 262 284 301 equa on239 identity 239 zeta function 244 Riesz F 291 Riesz Fischer theorem 291 Ritt J F 5 328 359 Robbins H 145 284 Rodrigues Olinde 340 Rodrigues39 formula 340 for Hermite polynomials 216 for Legendre polynomials 340 Rogosinski W 260 Runge Carl 569 Runge lutta methods 569 Saddle point 449 Sansone G 469 Sarton George 525 Schrodinger Erwin 219 503 wave equation 219 wave functions 219 Schuster M L 496 Schwarz H A 284 inequality 282 Scribner Charles J r 496 Second law Kepler s 117 Second law of motion Newton39s 116 Second order linear equation 81 Second order reaction 26 Section conic 119 Seeley R T 320 Selfadjoint equations 325 330 Separable equations 5 Separation constant 368 374 Separation theorem Sturm 158 leparation of variables method of 304 314 317318 368374 eparatrix 484 equence complete 290 orthomormal 278 323 equence of functions orthogonal 277 cries Bessel 361 binomial 175 Chebyshev 233 convergent 167 Fourier 249 279 301 cosine 268 sine 268 307 FourierBessel 361 Frobenius 188 Hermite 218 hypergeometric 200 INDEX 627 Legendre 344 power 167 sum of 167 Taylor 171 Shift rule exponential 134 Shifting formula 392 Simmons George F 249 262 275 407 Simple critical point 472 Simple discontinuity 258 Simple harmonic vibrations 107 Simpsoni s rule 570 Sine Euler s in nite product for 271 Sine series Fourier 268 307 Singlestep methods 558 Singular point 184 irregular 185 regular 185 Singular Sturm Liouville problem 334 Smith D E 223 242 Smooth function piecewise 296 Snell Willebrord 37 law of refraction 37 Solution general 8 84 88 linearly dependent 424 linearly independent 424 particular 8 periodic 486 trivial 84 422 Space metric 283 Special functions 166 Special Riccati equation 364 Special theory of relativity Einstein s 79 Specific heat 312 Spherical Bessel functions 359 Spiral 451 Spring linear 480 nonlinear hard 485 soft 485 Stable critical point 453 Standard form differential equation 159 Standing wave 311 Stationary function 508 Stationary value 508 Steadystate 112 315 Steinmetz Charles Proteus 114 Step function unit 388 Stephens E 135 Stoker J J 481 String stretched 298 struck 311 vibrating 303 Sturm J C F 158 comparison theorem 162 separation theorem 158 Stuarm LiuIwiIle I qTLIsaaI iDl39m pan5IiJnn prinibilemq P P P P siLnguIar 32 3534 S uDvessi w upprunimalium3 Piu39ardJ sV mathad at 541 Si39u3pemagtsiIiinnw p rinipI UL WE SysImA aut mnmuus au1iiIia r3r equmicrn nf 456 mnsamtive dyrnamiaal 436 Iiimar hamnugneaus 421 linear in h m g n U 421 umcnupld 43711 Szn aggyr Elm am A211 301 52136 6 520 Tanggm Lambr1quotsV cmn39tinu1edl frauti1m1 W fiIJr T39ME39EhJW39 mE 43 Tauttachmn pro3biemj 653 4103 V I4 Tauampt cihmnE Aprfnpertr Taylar sariegs W 1 aylr sEarm39ula 1770 Tiayinar rnethmJEs m HiirEI 1 563 Terrmirmaj valanitr ML Z Test w ltmpariquotn n ramiaA fl A1 TTlEEII39fj39 at par 11iutim1s I41 T11Ir1r anvf mlativity Einsteinquot 5perifaL T9 Ihr1mamndiuc tivi39t3a 312 r law He piErquot5 l Iietzci H Pp 7iIwhmarsLhi 245 E1 iTuKpIiu G2 6y Tpns1rwgy39 MU M575 TVmrielIi Eviangelista 3 law w mhgarem 43 Total diC lTlIE IifJ Ln1 r1 ar s Tnta relaiiwn trmr 5M mcn4 Ev Trajeclury 443 39Tran5nnd enrtaHfuntim1s highar 166 elcm eLmtaw 165 TransendenlaI numbers k Traxzsfurrm 3932 i n39 mac La P H3FErEE 392 iLaplacci 333 139ran5fnrmaIinn 332 imggra l 332 irwge rse LapIac Laplace 332 linsear 35 Tran5ient 112 T39ringl in quailit3F 2amp3 Triscalmi Pvb p c 1 441 4 Trivial snluti nn f d T1ruAvsed1ullK o 146 SW4 Twaadimensi nnal Lagpvhcz equamzinam N Twaedi mE nsiunal w ave aqguatium SW3 Ulam StazL1i5law 4111 UncaupId swarm 43 Umdamped pcndu Eum 4312 Unda3n1pg1 39aFihrati IT vibr timn 1m Undemminend 51iEnI TLJ ni Em39111mrwr gen e AUHEI VimgpuI5 vumtiIan 3amp9 Uni mp 4n rIium EH3 Unim2n EuIer 5 aittiaiunilE loward 145 Jeans div mitim1 on 39L Insrables17IivaiII point 54 1an at PDL Bal39Ihasar 439T 492 q1uatin jLilt 445 4 92 494 van der WagardEn Lit 515 0 mass 8EU quota able5 mesthoti mi 5E F T iI i39C39f i HE 304 3 IT 3TI3 p 34 Iquot39aria1 i4m1z of pa ram etars M4 434 far Iinear equaIi1ms i I3 lz a faI Alinear ampsrsten15 431 VEiI39UquotH39i IL 51 VEclurr5 imrre r praduaclz 280 numi Df D hDg aL Valacityz gscapc 5 trminaL 33 Vibrating mmbran 3 T392 339T FibraIing s1riing 0 sftratdied 293 struclz 3111 1J39ihra1iun 39GI39ili My dampcd 13910 l Eined H1 ifrEB I1 I vgrdamped 109 simkpi e harmmmic quotIE undamped I0 undardamped WIJ Avitzar uf EHrajf3r 413 V ilf 113T Volterra Vito 435 Volterra s preypredator equations 436 Vortex 449 Wallis John 139 152 Waltershausen W 8 von 221 Watson G N 77 240 348 356 359 361 Wave equation 3 366 536 onedimensional 298 304 Bernoulli s solution 307 d Alcmbert s solution 309 Schr6dinger s 219 twodirnensional 373 Wave function Schrodinger 219 Wave standing 311 Weierstrass Karl 247 284 INDEX 629 Weight function for orthogonal sequence 323 Westfall Richard S 153 Whewell William 150 Whittaker E T 240 Wilkes J 0 570 Wren Sir Christopher 43 148 Wren s theorem 43 Wright E M 142 Wronski Ho n 88 Wronskian 88 423 426 Zabusky N J 557 Zero of a function 87 Zeros of Bessel functions 360 Zeta function Riemann s 244 Zeuner F E 23

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