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finalrv2solns-2 - M427K Differential Equations 58045 2nd...

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Unformatted text preview: M427K Differential Equations 58045 2nd Final Review Thursday 3rd May 2007 INSTRUCTIONS 0 You have 3 hours. 0 Calculators are not allowed. 0 There are 9 questions, each worth 10 points. 0 Write your solutions on the question sheets, in the spaces provided. Indicate your final answers clearly by drawing a box around them. You must Show your working in order to get full marks. It is strongly recommended that you work out your solutions on scratch paper before writing on the question sheet. FOR INSTRUCTOR’S USE ONLY .1: .2: .3: .4: .5: .6: .7: .8: M427K Final Exam May 12th, 2006 Question 1 Find the solution of the initial value problem y’+ty=t7 y(O)=2- I.» I ' we“ firm? '9 jig/(k {is XH) -: e "- 6 M427K Final Exam May 12th, 2006 Question 2 [10 Points] Find the solution of the initial value problem dy_3x2—l d1}.— 4+2y’ explicitly. S06) (ma/4M6 W wllw M427K Final Exam May 12th, 2006 —-——-———————__—_____——_—__—_—_ Question 3 [10 Points] Find the general solution of the ordinary differential equation (3102.7; + 2mg + 3/3) dx + (x2 + yg) dy You do not need to find the solution explicitly. AA {Aka/W‘va $1041} JUL Do ~+ Wk 7, 2x S3 ,\ : €31 Mpg C. BDLLQ3X7 + szhkj l 7323ki>dlx + ()LIKSKA 7189967 :0 ‘1 Mme/ii w<117> ‘3: U’W’D‘l rs «x/l/w éol’vfiba M EL 2 3* t 3 3,: 2 mleshrfic“ 50 LP : 127e Jr $72 +C 1 Mlt D“? : glLe/l’fi] {. Zoroaskl. 7363L+L|(X)‘ 53L \ M—i Ct(1):o cfi/L CLL So 44!: Mlfilfim ‘3 M427K Final Exam May 12th7 2006 Question 4 [10 Points] Consider the autonomous ordinary differential equation Where f(y) = 2(y —- 2)3(y + 1)2.(y + 3)- dy_ E—fly) 1. [4 Points] Sketch f (y) vs y. S93 2. [4 Points] Find the equilibrium solutions and classify each according to its stability. r ‘ E: LLV‘N 64-0/3» sub“) 7 t: " 3 \7 : V l (,4le 1 * ' l 3. [2 Points] Sketch y(t) vs t. I r M427K Final Exam May 12th, 2006 ————_________________________ Question 5 [10 Points] 1. Find the general solution of the inhomogeneous second order ordinary differ- ential equation 3/” — 2y’ ~ 33/ = (—815 + 6)e"t. ., _. \ v‘ ‘ val-«bx 32/3 : Cmpz)(~+\>vo 4) M~~3J 3f: ’ —t_ 80 7L: g, 6 leve CM) 7', ; @131 gr) a“ w ’ 7’; ; (,AtLJFQR—{ék + @4216 . wt w“=<1\u + H—[email protected]>+r + WWW ' ma V [+ we e ~~ A + w —— 1 16+: 3r CY/A’ufi +i53> : C'Xtfiq’ so: +8A>~WY$= + ) lee-463E Zj')HA:<6,A-:1 1, + [3:2 ) :3) K:‘- 2. Find the complementary function and a suitable form for a particular solution of y" —— 23/ + 2y = at + at cos(t) + sin(t). You do not need to find the undetermined coefficients. M427K Final Exam May 12th, 2006 Question 6 [10 Points] Find the general solution of the ordinary differential equation xzy" + 5mg + 4y = 0 given that 311(av)=x_2 is a solution. i c: V be (‘M 7 t: V I :1 Al 7‘ZV' ’zu 7":V «LB'J‘ £3 /‘ w.— "—‘ 2 ‘1 )3 “’3 J 9&1 3; } M427K Final Exam May 12th, 2006 Question 7 [10 Points] Find the general solution of the ordinary differential equation y”+$y’+2y=0 about the point x0 = 0 by using power series methods. You should find the recurrence relation and formula for the general term if possible. c4 «Q 3-— —— A 1“) \r ;, “(L h“ 7 l :1 Z awhmm '|) 7w: L X441<m4ly M) A: J ‘ h: I “ti; Sks’bShV-vl’w‘hlzwo .\/\ C I: i“ 3%)“ (4% [BLAJAkLA + Ila/LA'I + Qééxnv :0. 4:1) +1, my, oil .2 l + Q «C, i 00 : g x + /\0\ “hack J , L L th (n+2) (,W‘Ll) 4 A: | A Mr: or L : - or.) “H mm “MW—M “ m N M/'/ a : r L 01A {gr 4 2 l 4+1 wfiflfl Ow A A i l , a : Il’f a! b Q/ ZIL-l—i It; 7" ’l \ ' A. ‘1’“ M427K Final Exam May 12th, 2006 Question 8 [10 Points] Solve the initial value problem y” + 23/ + 22/ = 9(t), y(0) = 0, y’(0) = 0 where 5sin(t) 0<t<7r t: _ g() {0 fig using the partial fractions expansion 5 _ 23+3 23—1 (52+1)(32+2s+2) _s2+25+2 s2+1 (l- MW .5gsl/Lt = 56;le T HWL+>SQFA Question 9 [10 marks] Consider the periodic function defined by f(96)=Ilc for —1 S a: < 1, with f (x + 2) = f for all :1). Use the Euler—Fourier formulae to calculate the Fourier coefficients of f. ...
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