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# EL DIFFERENTIAL EQ MATH 2065

LSU

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This 12 page Class Notes was uploaded by Madison Gottlieb Sr. on Tuesday October 13, 2015. The Class Notes belongs to MATH 2065 at Louisiana State University taught by William Adkins in Fall. Since its upload, it has received 45 views. For similar materials see /class/222634/math-2065-louisiana-state-university in Mathematics (M) at Louisiana State University.

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Date Created: 10/13/15

Math 2065 Section 1 Final Exam Review Sheet The nal exam will be on Wednesday May 12 from 1000 AM to 1200 Noon in the normal classroom The exam is closed book but you will be provided with the usual table of Laplace transforms The nal exam is comprehensive and thus any of the material we covered is a valid source for questions You should collect each of the review sheets and exams as sources for study for the nal exam Each of these is posted on the class web site in case you have misplaced them A good strategy for study is to do the review sheets and exams without looking at the answers If you then compare with the answer sheets you can identify the areas in which you need additional work Some additional review exercises are included here of exactly the same type as found in your text and on the previous review sheets Review Exercises Solve each of the following differential equations 1 y t 7 2y 2 y t 7 4ty 3 y ily t4 t 4 yy ti 12 5 y1ty2ty2 6 For the equation y 17t y a Find the general solution b Find the particular solution with y2 1 and give its interval of existence 7 Consider the initial value problem 253 5t 7 y y1 e a Without solving the equation give the domain of existence of the solution as guar anteed by the existence and uniqueness theorem b Now solve the equation and see if your answer is indeed de ned on the interval you found in Part a 8 y 73y2y0 to y 2y 2y0 H 0 y 4y 4y0 y 6y 9y0 y 7 62 13y 0 y 16y 0 5224931 07 240 07 MW 1 y SQ139 07 240 7 MW 1 y 2y y0 y 2y715y0 tzy Qty 7 6y 0 3t2yH11ty73y 0 tzy Qty173 0 t2 73ty4y0 39y72yy352t y 21 y 2equot y 7 y 7 2y 79571 et y 2y yti5 1 1 y 11 7 723 ln 257 t gt 0 You may assume that a fundamental set for the associated homogeneous equation is 90105 t7 47205 t l the Laplace transform of each of the following functions 39 t2679t 52 7 t3 t2 7 sin 5t tcos6t 2sint3cos2t e 5t sin 6t t2 cos at where a is a constant 1 if 0 g 0 lt 27 33 ft 71 if 2 g t lt 47 and 0 if t Z 4 34 ft t2 7 100ht 7 10 Find the inverse Laplace transform of each of the following functions 4 39 52 7105 9 2s 7 18 52 9 2s 18 39 52 25 s 3 39 52 5 s 7 3 52 7 6s 25 1 40 7 552 4 525 12 1 7 5 9 8 35 39 41 42 1e77rs 43 7 52 1 1 72 44 LetAi73 2 a Compute 517 A and 517 A 1 b Find 1517 A 1 71 131 c What is 5quot i i i i 3 71 Solve the matrix differential equation y Ay Where A 75 71 d Solve the system y Ay7 y0 4 U 46 Solve the initial value problem y i y y0 Ol 47 Solve the initial value problem y g 763 y y0 48 Consider a pond with 1000 cubic meters of water There is a stream owing out from the pond at a rate of 10 cubic meters a day Nearby is a eld which is regularly irrigated and fertilized Each day7 10 cubic meters of water from the eld enters the pond7 and this is contaminated with 3 kilograms of ammonium nitrate per cubic meter Write down a differential equation for the amount of ammonium nitrate in the pond at time t Assume the ammonium nitrate is perfectly mixed and ignore the effect of rain and evaporation Do not solve the equation H 9 9 wa y c1672t C2te y 01 3t C2te T Answers y ti icei2t 4y 1ce 2t2 y its 312 7 2t7 13 c arctanyit7c a y b y m St a 07 00 b W 7 y alet 826 The interval of existence is 17 0 y e tcl cost 02 sint 72 is y ale3t cos 2t age3t sith y cl cos 4t 02 sin 4t y ac 7 at y 673003 2t si112t y Clelt1 gtt 0261ix5 y Cle t C2675 y clti3 02t2 y cltl3 Cgtia y t 4cl cosln 02 sinln t y 8272 8272 In M y alet Cgtet 3e2t y cleft Cgtei t t2e t y cleft age2t 3te t y alet Cgtet t aet 71 2 4 2 yc1tczt Lnt7 t 2 59g 5 2 7 572 s7 E s225 2736 52362 2 35 s21 s24 6 s536 2s345sa2 524M2 172572s5745 257105 2057105 s2 w c 44 45 46 whet 20033t76sin3t 20035t15783in5t cosxgtsingt e3t cos 4t i icoth te t2e tt72 17ht71 sint17ht77r s 72 7 571 2 717 574s1 574s1 a 5I7A7 3 5725I7A 7 73 571 574s 1 574s1 2 3 t 72 2 t b E igeu i Ct 734t87t C 6At is same as Z 15I7 A 1 7 1 86 36 t d 3 i g 216 7 6e4t t 7 l 501 i 02 C1 cle 2t y 7 6 7501 02 502 501e 2t 1 4t yt 72 6264 3t3t63t 3 7263 7 3763 If yt denotes the number of kilograms of ammonium nitrate at time t then yt 7 t 7 307 Exam III Review Sheet Math 2065 The syllabus for Exam III is Sections 367 41747 467 517557 the matrix algebra supplement7 and 71773 You should review the assigned exercises in these sections Following is a brief list not necessarily complete of terms7 skills7 and formulas with which you should be familiar 0 Know how to solve the Cauchy Euler equation atzy My cy 0 where a7 b7 and c are real constants by means of the roots r1 p2 of the indicial equation qr arpi 1brc0 See Theorem 47 Page 227 0 If ltg1t7 y2t is a fundamental solution set for the homogeneous equation 2 btz Cty 0 know how to use the method of uariation of parameters to nd a particular solution pl of the non homogeneous equation 24 WW Cty W In this method7 it is not necessary for ft to be an exponential polynomial Variation of Parameters Find pp in the form yp ulyl ugyz Where u1 and IQ are unknown functions whose derivatives satisfy the following two equations 9 uiyl l 71292 0 MM U 2y 2 1 05 Solve the system for u 1 and u 27 and then integrate to nd u1 and ug 0 Know what it means for a function to have a jump discontinuity and to be piecewise contin UOUS 0 Know how to piece together solutions on different intervals to produce a solution of one of the initial value problems I y ay t We yo or y W by t We 2407 2050 2417 where f t is a piecewise continuous function on an interval containing to 0 Know what the unit step function also called the Heauiside function ht 7 3 and the on o switches Mam are 0 if0gtltc ht 7 c hct 1 if C gtt and7 1 if a lt t lt b i h t7 7 h t7 b MW 0 otherwise a 7 and know how to use these two functions to rewrite a piecewise continuous function in a manner which is convenient for computation of Laplace transforms Exam III Review Sheet Math 2065 0 Know the second translation principle Theorem 8 page 270 C 7 cht 7 0 e csFs and how to use it particulary in the form of Corollary 9 page 271 C gtht 7 0 e cs gt 0 as a tool for calculating the Laplace transform of piecewise continuous functions 0 Know how to use the second translation principle to compute the inverse Laplace transform of functions of the form e Fs 0 Know how to use the skills in the previous two items in conjunction with the formula for the Laplace transform of a derivative Theorem 3 Page 278 and Corollary 5 Page 280 to solve differential equations of the form 2 ay 1 05 240 yo and y a2 by t 240 2407 20 2417 where ft is a piecewise continuous forcing function 0 Know the formula for the Laplace transform of the Dirac delta function 6025 C 60t 5 Know how to use this formula to solve differential equations with impulsive forcing function 2 W KM 240 yo 2 a2 by K6005 240 2407 MW 241 o For a constant coef cient homogeneous linear system 2 Ay 110 907 the unique solution is W eAtym where the matrix exponential 5quot is de ned by the in nite series At 1 2 2 1 3 3 1 n n e InAt7At 7142 7At 2 3 nl See Theorem 723 page 383 and Corollary 739 page 395 Know how to compute 5quot from the de nition for some simple 2 gtlt 2 matrices as is done in the examples on Pages 3917392 Exam III Review Sheet Math 2065 0 You should know how to do the algebraic operations on 2 x 2 matrices Addition7 scalar multiplication and multiplication of matrices is reviewed in the matrix algebra supplement and in Section 61 The following two facts for 2 x 2 matrices are fundamental IfA Z 1 then detA ad 7 bd 7 0 ltgt A is invertible and 1 d 7b th 0th 1447 1 e 7 en ad7bdl7c al 0 If A Z 7 then the system of differential equations 2 Ay 110 Cl 7 C2 which can also be written as 241 ay1by2 0 0 22 Cy1dy2 y1 017 y2 62 has the solution 1105 5mm Since eAt 715I 7 Arl7 see Corollary 77 Page 394 the following algorithm can be used to solve y Ay when the matrix A is a constant matrix Algorithm 1 1 Form the matrix 5 7 A 877 8de 2 Compute the characteristic polynomial 195 detsI7 A det 877 8de 52 7 a 15 ad 7 bd 3 Compute s 7 d b 1 s 7 d b 7 717 7 p8 298 1 m malc s7d 8 198 195 4 Compute 7 d b 3 f h t h2t 1 7A 1 7 195 195 1 8 71L 71s7a 130 h4t l M maJ 5 The solution yt is then yt 7 wwm7w4ud7m49wm7V mlflt gli gl Exam III Review Sheet Math 2065 The following is a small set of exercises of types identical to those already assigned Solve each of the following Cauchy Euler differential equations H 2937121 0 d tZyHty16y 0 e 425 7 7ty 6y 0 f tzy my 4y 0 Find a solution of each of the following differential equations by using the method of variation of parameters In each case7 5 denotes a fundamental set of solutions of the associated E0 homogeneous equation a y 23 y tile t S e47 te t b y 9y 9sec3t S cos 3t7 sin 325 c tzy Qty 7 6y t2 5 22 3 d yu 42 4y Isl25721 S 67217 t572t 3 Graph each of the following piecewise continuous functions 2 if 0 lt t lt 17 t 7 a H 3722f ift 1 b 1 05 6 2quot1ht 7 1 C W tX02 7 3Xl27 4 t 7 42ht 7 4 4 Compute the Laplace transform of each of the following functions a t t2ht 7 2 X24 b N t 1 t if 0 g t lt 17 C t t2t71 if1tlt2 72 if t Z 2 5 Compute the inverse Laplace transform of each of the following functions 1 7 5 9 a M 5 1 e77rs b F 7 7 lt gt s 82 1 86725 a 1 82 7 9 d Fltsgt7lt1e7 2gt5 1 52 25 10 Exam III Review Sheet Math 2065 6 Solve each of the following differential equations 2 if 0 lt t lt 2 a 5 7 0 O Hyy t1ift22 y b y 4y t1ht27 240 07 MW 1 C y 4y 462037 240 717201 1 72 LetA73 2 1 a Compute 517 A and 517 A 1 Find 4 in A 1 c What is 5 d Solve the system y Ay7 y0 00 Solve the matrix differential equation y Ay where A E5 to Solve the initial value problem H O U 2 lt D d 3 CD EC 9 9 4i C D U is O E D 5 Exam III Review Sheet N F 5 03 77 00 50 Math 2065 Answers a y m3 c2t5 b y t32c1 c2lnM c y m4 c2t 3 d y 51 cos4ln C2 Sill41nw e y clt m C2t6 f y r2c1 c2lnM a mt t In tequot b ypt n cos 3t cos3t 375511137 C Mt 15t21n t d My 415t52 72t a Fs e S wwwww gwwltgg lt F 7 af w4f g9 a 1 1ht1 b ft 1 7 ht 7 7r sint C 1 63quot1e 3quot1ht 1 d equot cos3t 7 s1n3t e O WQ cos 3t 7 7r2 7 gsin3t 7 7r2 ht 7 7r2 a W 17 e 5 650 5t7 2 7 6 W7 2 b yt sin2t ht7 2 t7 2 3 7 3cos2t 7 2 7 sin2t 7 2 c yt sin2t 7 cos 2t 2ht 7 2 sin2t 7 2 s 7 2 72 wsI7Afg1 34I7Ar47 9 1 f i 7 D 7 D b tgenfgt ling c EA issalneas 571517 A71 7864 36quot ym7 m wJ 551 7 62 4t cl cle 2 7 l ya 7 5 7501 62 4t 502 501e 2 1 64 7 l yagt7222 1 63 3t 3t 305 7263 7 3763

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