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- StudyBlue
- Michigan
- University of Michigan - Ann Arbor
- Mathematics
- Mathematics 216
- Epstein
- Fall 2002 Final Exam

Anonymous

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Name: Section/Time of lecture: Professor/GSI: December 13, 8-10 am, Aud C, Angell Hall Each part of a problem counts equally. To get full score you need to carefully explain what you did. 1 Problem Points Score 1 8 2 8 3 8 4 8 + 3 3 TOTAL 35 2 Problem 1. a) Verify that y = ex −e−x satisfies the differential equation y′ = y + 2e−x. b) Solve Solve dydx = x√x2 + 9,y(−4) = 0. c) Solve dydx = (8xy)13. d) Solve (x2 + 4)y′ + 3xy = x,y(0) = 2. 3 Problem 2. a) Solve y′′ + 8y′ + 25y = 0. b) Find a particular solution of 2y′′ + 4y′ + 7y = 3x2. c) Find the matrix products AB and BA if they are meaningful. d) Solve x′ = 7x−5y y′ = 4x + 3y 4 Problem 3. a) Decide on stability for the critical point (0,0): x′ = y y′ = −x b) Find all critical points and decide stability: x′ = x + x2 + xy y′ = y −y2 −xy c) Calculate the Laplace transform of the function: d) Solve x′′ + 2x + 4y = 0 y′′ + x + 2y = 0 x(0) = 0 y(0) = 0 x′(0) = −1 y′(0) = −1 5 Problem 4. Include table a) Recall that L(eatf(t) = F(s−a) and L(tn) = n!/sn+1. Find Lf if f = e5tt4. b) Let f(t) = t and g(t) = t−5. Find f ∗g. c) Recall that L(u(t−a)f(t−a)) = e−asF(s). Find f if L(f) = e−3ss2 . d) Solve x′′ + 4x′ + 5x = δ(t−pi) + δ(t−2pi) x(0) = 0 x′(0) = 1. 6 finalf02.dvi

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