Fall 2002 Final Exam

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 = ?(t?pi) + ?(t?2pi) x(0) = 0 x?(0) = 1. 6 finalf02.dvi