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# Class Note for MATH 3331 with Professor He at UH

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COURSE
PROF.
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TYPE
Class Notes
PAGES
37
WORDS
KARMA
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This 37 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 15 views.

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Date Created: 02/06/15
Lecture 20 s 3 m MVEVSE Lamace mmmm x 1W He Wm M WM WW M mm quotA iivlavamZXmm U niqueness Theorem Thm If ft and gt are piece wise continuous on 0 g t lt 00 and fs gs for s gt a then ft gt for all t in O s t lt 00 at which ft is continuous Math 3331 Section 19470 Lecture 20 March 11 2009 2 21 Section 53 C Transform Pairs Fs fs fts s gt a L transform pairs 0 ft determines Fs uniquely in s gt a o F3 determines ft uniquely in 0 g t lt 00 except at discontinu ity points Jiwen He University of Houston Math 3331 Section 19470 Lecture 20 March 11 2009 3 21 Def Given Fs and ft st Fs fs then ft is called the inverse Laplace 1 transform of Fs and is denoted by W 1Fgtt 1Fltstgt F 2 gm lt gt r 1F March 11 2009 4 21 Math 3331 Section 19470 Lecture 20 Table of Inverse Transform r F f ltgt f L 1F Fltsgt L1Fltsgtlttgt L ed 8 0 1 75k 1 ect 5 0 ls 1 1 eatsin 6t s a2 2 Wig 21W eat cos t I Jiwen He University of Houston Math 3331 Sermon 19470 Lecture 20 March 11 2009 521 Section 53 Basic Definition Ira Class Exercises Partial Fractions Examles Exercise 532 2 Compute the inverse Laplace transform of Ys 335 2 Adjust as follows y 2 2 1 S 3 5s 5 s 35 Thus by linearity 2 1 x 3 yo i 5 s 35i 251 1 5 s 35 2 35t 53 Iill Jiwen He University of Houston Math 3331 Section 19470 Lecture 20 March 11 2009 6 21 Section 53 Exercise 534 4 Compute the inverse Laplace transform of Ys fj g 4 Adjust as follows 5S 3 Y 5 S S2 9 52 9 Thus by linearity s t 1 1 539 s2 9 s 5 1 99 250053 Ill Jiwen He University of Houston Math 3331 Seetion 19470 Lect LJ re 20 March 11 2009 7 21 Section 53 Basic Definition In Clas Exercise Partial Fractions Exam les Exercise 536 6 Compute the inverse Laplace transform of Ys g 2 6 Adjust as follows 2 1 3 Y S 354 9 94 Thus by linearity 1 3 yl 1 3 l 9 s4 Jiwen He University of Houston Math 3331 Section 19470 Lecture 20 March 11 2009 8 2 1 Section 53 Basic Definition IneClass Exercises Partial Fractions Examles Exercise 537 7 Compute the inverse Laplace transform of Ys 5325225 7 Adjust as follows 3s2 Ji S sz25 3s 2 3225 s225 3 5 2 5 s225 5 s22539 Thus 3 2 5 t 1 yo s2255 s2zsi s 2 5 35 1 c1 is2zsi5 5225 2 30055t 5 sin5t Ill Jiwen He University of Houston Math 3331 Section 19470 Lecture 20 March 11 2009 9 21 Exercise 539 Section 53 Basic Definition ln Class Exe39rcis Partial Fractions Exam les 9 Compute the inverse Laplace transform of Ys 3 25 24 49 9 Adjust as follows YS 1 3 2s 3 4s s249 1 1 3 2s 4 s 34 s249 s249 1 1 3 7 4 s 34 7 s249 S 2 s249 Thus 1 1 3 7 1 y 4 s 347 s249 S 2 s249 1 1 3 7 L 4 s 347 is249i S 2 3249 1 3 34 sin 7t 2cos 7t 46 7 March 11 2009 1021 Math 3331 Section 1940 Lecture 20 Jiwen He University of Houston Section 53 Exercise 5311 11 Compute the inverse Laplace transform of Ys L 11 Note the transform pair By Proposition 212 2 s2339 1 5 E s23 15 2 39S23 i 2 523 6 2 t2 H e ZttZ Thus d A N V H NIUI N101 h Jiwen He University of Houston Math 3331 Section 19470 Lecture 20 March 11 2009 11 21 Section 53 Basic Definition Exercise 5314 45 1 14 Compute the inverse Laplace transform of Ys m S By Proposition 212 1 et COSZI 4 a 1V4 Hence 4s 1 131 m y s 124 1 s l 4 ia 1 4i 4e39 cos 22 m Jiwen He University of Houston Math 3331 Section 19470 Lecture 20 March 171 12 Section 53 Exercise 5315 39 25 3 15 Compute the Inverse Laplace transform of Ys S125 15 Note the transform pairs ThUS t 1 25 3 cos tltgt S y 3 125 325 2s 2 1 5 1 sinxgtltgt Sl25 s125i s2 5 S 1 5 1 7 By Proposition 212 s 1 5 1 f5 39 T 5 deck51 6 s s 125 2 s l J5 f5 quot s 12 5 T m s ALA i 3 s 125 1 2 2e39 cos t e sin xgt d5 e 2 cos xgt g sin 50 Jiwen He University of Houston Math 3331 Section 1947 Lecture 20 March 11 2009 13 21 Section 53 Basic Definition In Ciass Exercises Partial Fractions Exam les Exercise 5317 39 35 1 2 17 Compute the Inverse Laplace transform of Ys S24S29 17 Complete the square Thus Y 1 S s s24s29 s2225 W s2225 Note the transform pairs g 4 3 s 6 4 055 s s2225 s2225 s2 25 1 s 2 sin5t ltgt s2225 S2 25 4 5 By Proposition 212 3 I S 22 25 2 2t s2 351 e cosStltgtlts2gt225 s2225 5 4 5 2t t 1 8 ms ltgts2225 5 s2225 4 3e 2 cos 5r 562 sin 51 4 e392 3 cos 5 3 sin 51 Jiwen He University of Houston Math 3331 Sedion 19470 Lecture 20 March 11 2009 14 21 Section 53 Basic Definition In Class Exercises Partial Fractions Exam les Inverse L Transform of Rational Functions Inverse L Transform of Partial FraF FiOH Rational Functions DecomDOS39tIOH PHD Form Fs 2 138 178 Z FMS 628 A o P3 Qs polynomials degree Of P lt degree of Q 1435 contribution from root A Linearity gt time Z Home A Assume Qs has is distinct roots 39Let m be the multiplicity of A Set 653 2 Qss quotm gt QAO 7k 0 Ilil Jiwen He University of Houston Math 3331 Section 19470 Lecture 20 March 11 2009 15 21 Section 53 Basic Definition ln Class Exercises Partial Fractions Examles Simple Root m 1 Simple Root m 1 A PO 5 a A QT gt 1Ft A6 gomplex Case Assume A a 213 Real version let A a ib A 2 Oz 7L5 are a complex conjugate 2a8 a 2195 pair of simple roots gt FMS l FX5 8 12 52 A Z gt FA5FX5SX gt 1F I FXt gt 1FAFXt AektZ Xt 2eataCOS t bsin t 2ReAegt t Illl Jiwen He University of Houston Math 3331 Section 19470 Lecture 20 March 11 2009 16 21 Section 53 Multiple Root m gt 1 Multiple Root m gt 1 Am Am l A2 A1 Coeffncnents d 1 1 9 P 8 S A m l s A m lt A ld8j1QSlsA 2 L 1ltFltsgt eAtlAmAm1t A1tm1m 1 For multiple complex pairs A For m 2 1FFt A POO A i 135 2 ReAm t tReAm1 t 1 C27 2 d8 QASl8 Ltin 2ReA2le 5 tm lReA1tet m 2 m 1 Illl Jiwen He University of Houston Math 3331 Section 19470 Lecture 20 March 11 2009 17 21 quotE39 qws Mum1m T V Lecture 7 5 Szcimn 21 Laamw y Lu My 1W5 H m Lam f apmmmmmmsu meuam quotA divlavamZ mm Wm quotmay meh grlv znu Frm Differential Forrnsand Diferential Equations F Marina 3W dy PIy dw Qmy Section 26 39 ifquot or Pmyowy 0 lt1 0 I Differential form formulation an 2 Pltwygt 2im oltmydyo lt2 huh 4 l J whd39cs off a d F Q mama IIii Math 3331 Section 19470 Lecture 7 February 4 2009 2 18 Form Section 26 a h Solution Curves and Integra Curves I 39 igiquot quot quotw quot x 39 e I ex F095 7 3 we 2 C 7 W Jsc Level cur es of a function F 39hr 0 o r r X e F 2 Form 3 Z py J 7 g 0 lt3 7quot Qty On curves 3 d dy ll P 7 7 Z O 1 F F or my Qltmydm ltgt dFEadadyO 4 Differential form formulation 32 8y o Curves 3 are integral PCB ydI 62137 ydy 0 2 9E curves for 1 if 3 defines quot3 T 80 a 33 ImpICIt SOiUtIOfiSOf 1 3 CW 2 la13llalrw calw Fl Jiwen He University of Houston Math 3331 Section 19470 Lecture 397 February 4 2009 3 18 Section 26 39 Example 30 Q g s a y r U lt2 TN 3 C4 i gtJ39O39Jg EXi y ar 56 y c1 an Iquot jzjlydydaij r f x 3Q A m d offdswrydizo 37 my fad 511 ydyAC F x jyz xz39szj Integral Curves V 39quot39I39IquotHgtiv22y22 C ly Sal Cadav Integral curves are circles Kym fa Explicit solutions 3 t la ix C gt0 Z 3070 2 Sb CWW 933 Q 20 SC Fro Val Glow a Explicit Solutions 1 clt39o H cWV L Jiwen He University of Houston Math 3331 Section 19470 Lecture 7 February 4 2009 4 18 Section 26 Form uation 4 Mxi39 dj de 6M 3 dy P7ygtP L Level curves of a function F Form d 3 Fltmyo lt3 or PmyQmy 0 1 On curves 3 quotDifferential form39iormulation dFaFdaFd O 4 8w 3y y dF Pltccygtda QIy dy39 0 2 DE t 1 0 2 is exact if existsf st 39 If 2 is exact then 3 P 6F3at Q 8Fay defines integral curves la 2 a 3 3 r H Qquj 3939 Q Pfx EJ 3 Ex tfaa xfr39 i a y0is exact lt2 Fxyx2 y22 i quotF him 3 dx L2 j dy 0 is exact lt2 Fxy y42 To 3 991L 391quotz 17C Fiji39s 6222215le ill 932 2c X amp J Jiwen He University of Houston Math 3331 Section 19470 Lecture 7 February 4 2009 Cna z leeniu fuvm u 3 de Qdy MW dc m m n n 5 mu d r 2 2deerem2Homv mm sthezeznsasywzywimdf quot m m at 4 Q 4w Cna z leeniu fuvm u 3 de Qdy MW dc m m n n 5 mu d r 2 2deerem2Homv mm sthezeznsasywzywimdf quot m m at 4 Q 4w Section 26 i ii Condition for Exactness 4 l dy Pcy Condition for exactness Form 13 7 M y 8P 82F 8Q a F 8y Bm y 81 813811 or PltxygtQltmygt 0 1 Differential form formulation VPCL39 yd FQC B ydy 0 2 3 0 5Axgf J o 2 is exact if exigts F st P 315690 Q aFay EX xdxydy0isexact lt2 gig 20 I 1 8y 8X P 9 ch sa p39cv EX xdx l 2y3dy20is exact Q 20 fx 6723 11 Jiwen He University of Houston Math 3331 Section 19470 Lecture 7 February 4 2009 7 18 Section 26 tions fwaJZfMjr Special Case separable Equa Special Case Separable Equations 9 QXM39T an dyP A J kr J oa 3 xfo m mw AF Pltmdm Qydy 0 gt M JkC PwdQydy0 2 gtfeJfPrxar fe714x 2 33L 5 J I Math 3331 Section 19470 Lecture 7 February 4 2009 818 Section 26 iv t Form M W dy i K W s DE Form ZJDWy Find F if 5 is satisfied d9 6206731 or pltygtQltmygt yo lt1 aFamP J J vDifferential form39igornfulation39 9PT H 7 39 gt39 319 y 3719 y Pltxygtdx39c2ltxygtdy 0 lt2 HOE f mm Cir P Q 3 U 2 7y 7y J o 2 is exact if exists F st pjgp an 39 8F892Q J14 f T 8 l i Conditio br exagcness gt y qb Qm y LID EM DE cp 771 3 8y 8568347 813 a y 3 amp 5 Z 54 Solve this for y Ber 8y 8 VL rns depen s only on y I a igqg Qr3239 62353quot Jiwen He University of Houston 1 Math 3331 Section 19470 Lecture 7 February 4 2009 9 18 Section 26 iv t Form M W dy i K W s DE Form ZJDWy Find F if 5 is satisfied d9 6206731 or pltygtQltmygt yo lt1 aFamP J J vDifferential form39igornfulation39 9PT H 7 39 gt39 319 y 3719 y Pltxygtdx39c2ltxygtdy 0 lt2 HOE f mm Cir P Q 3 U 2 7y 7y J o 2 is exact if exists F st pjgp an 39 8F892Q J14 f T 8 l i Conditio br exagcness gt y qb Qm y LID EM DE cp 771 3 8y 8568347 813 a y 3 amp 5 Z 54 Solve this for y Ber 8y 8 VL rns depen s only on y I a igqg Qr3239 62353quot Jiwen He University of Houston 1 Math 3331 Section 19470 Lecture 7 February 4 2009 9 18 Section 26 Example 96 m I 9 27 F 39 an 39 x sm e 46 az Pd EX eyxey sinyj i 1136 quot feydmzmei GOV 63daey sin dy0 0 CIDy quotQ 3H3yquot D5 P 9 w smyww Here PIy ey Siny a ofx 0137 meg Sing gt dgty cosyPC Check exactness gt F06y Hy y END89 698Q8m 63 fey cosy 7quot C Implicit So Equation is exact gt Find F W 98 P 3 33 a o 3F I 39 339 Hf fq 915 i jk39smj gt 4D imam of SA H Jiwen He University of Houston Math 3331 Section 19470 Lecture 7 February 4 2009 10 18 In Class Exercises EXercise 269 EX 9 233yd96 sv 6ydy O Pmy2ay Qym 6y 8P8yl 8Q3m1 gt exact Hltzvaygt Pcydcas2ccy My Q aHay ac 6y c 6y gt My 3y2 gt Fy Hwy CIDy 2 my 332 implicit solution 32 my 3y2 2 0 Av 3 W el Jiwen He University of Houston Math 3331 Section 19470 Lecture 7 February 4 2009 11 18 In Class Exercises Exercise 2611 EX 11 1yda 1ady O P 1ym Q 1r gt 8P8y2 190 BQsz 1522 gt not exact oi A yourSei Jiwen He University of Houston Math 3331 Section 19470 Lecture 7 February 4 2009 12 18 In Class Exercises i Exercise 2613 7 EX 13 dydm 3m2 y3y2 CC or 3m2 ydw a 3y2dy O P 32y Q 110 3312 gt 8P8y2 l 8628513 1 exact HPdcc3ry gt lQ 8H8ym 3y2 5y 3y2 gt Dz 3f FHdgtx3my y3 gt implicit solution w3xy y30 Jiwen He University of Houston Math 3331 Section 19470 Lecture 7 February 4 200g 13 18 i l Integrating Factor 39 i v Integrating Factor If PC7341 d3 QIy dy 0 is not exact try to find func tion u1y St HP dzz HQ dy O is exact Mx y integrating factor Jiwen He University of Houston Math 3331 Section 19470 Lecture 7 February 4 2009 14 18 Integrating Factor v Example EX m l ydc cdy0 3y9y 1 not 8 8 1 exact Use u 12 gt 1M yIBQMQS 133dy 0 81xyas2ay was 8 18 12 Find F H yx2d In y cp 1x aon icci mm o gt OgtFHn yx Fl exact February 4 2009 15 18 Jiwen He University of Houston Math 3331 Section 19470 Lecture 7 Integrating Factor o A function Gzy is homogeneous of degree n if G y X Gy o The DE 1390 3 d l Qy dy O is homogeneous if P Q are homogeneous of same degree a If the DE is homogeneous then the substitution y 332 transforms it to the separable DE d3 Q1vdv 3 P1 U volt1vgt 2 EXi Gmy 1562 22 gt GOwc Ay 1Aac2 AZ2 A 2cc2 22 gt Gxy A 2Gy gt homogeneous of degree 2 HEXI Gm y x3 va 5 0090 kg MP MOW2 A3933 W gt CAxy 3GUy gt homogeneous of degree 3 Jiwen He University of Houston Math 3331 Section 19470 Lecture 7 Februa 4 2009 16 18 Homogeneou In ClasS Exercis 2635 EX 35 x2y2dac 2cvydy 0 Sub yxv gt dyvdxacdv gt x2x2v2das 22vvdxdv0 gt 1 v2dac 2xvdvO gt dajac l 239Uv2 1Hdv20 nxnv2 1nv2 1D v2 1C CieD Sub vy7c gt y2 x220m Jiwen He University of Houston Math 3331 Section 19470 Lecture 7 February 4 2009 17 l8 n Ema 2 a 13 N m wzh

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