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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:
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.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“
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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 $1041} JUL
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Consider the autonomous ordinary differential equation Where f(y) = 2(y — 2)3(y + 1)2.(y + 3) dy_
E—ﬂy) 1. [4 Points] Sketch f (y) vs y. S93 2. [4 Points] Find the equilibrium solutions and classify each according to its stability. r ‘
E: LLV‘N 640/3» sub“) 7 t: " 3 \7 : V l (,4le 1 * ' l 3. [2 Points] Sketch y(t) vs t. I r M427K Final Exam May 12th, 2006
————_________________________ Question 5 [10 Points] 1. Find the general solution of the inhomogeneous second order ordinary differ
ential equation 3/” — 2y’ ~ 33/ = (—815 + 6)e"t. ., _. \ v‘ ‘
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[+ we e ~~ A + w —— 1 16+: 3r CY/A’uﬁ +i53> : C'Xtﬁq’ so: +8A>~WY$= + ) lee463E 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
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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“)
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' 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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 Spring '08
 Fonken
 Differential Equations, Equations

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