s251final(sp04) - Math 251 Spring 2004 Final Exam ANSW ERS...

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Unformatted text preview: Math 251 Spring 2004 Final Exam ANSW ERS 1. a) True; b) True; c) False; d) True; e) False; f) True 2. a) µ(t) = t−3 ; b) c = 3; c) Y (t) = Ate−2t + Bt2 + Ct + D. 3. a) T = √ 2π (sec); b) µ = 5 (rad/sec); c) It is critically damped. 3 4. y(t) = u1 (t)(t − 1)e3(t−1) − 2u3 (t)(t − 3)e3(t−3) 5. x′1 = x2 x′2 = −5 x1 + x2 2 or 1 3 e4t + C2 2 1 b) It is an unstable saddle point. c) α = −1 d) 6. a) x(t) = C1 0 1 1 x′ = −5 2 x e−t cos t − 4 sin t e3t 4 cos t + sin t b) It is an unstable spiral point. 7. a) x(t) = 8. a) The critical points are (0, −2), (1, 1), (−2, −2). b) (0, −2) is an unstable saddle point, (0, 0) is an unstable spiral point, and (−2, −2) is an asymptotically stable node. 9. λX ′′ − X ′ + 5x3 X = 0 λT ′ − T = 0 n2 nx 1 4 9 , n= 10. The eigenvalues are λ = . , , ..., , ... Their eigenfunctions are yn (x) = Cn cos 4 4 4 4 2 1, 2, 3, 4, ... b) Yes, λ = 0 is an eigenvalue. Any nonzero constant function is a corresponding eigenfunction. 1 −1 1 mπx mπx 1 (−2 − x) cos dx + dx + x cos 2 −2 2 2 −1 Just state that am = 0 for all m (by noting that f (x) is an odd function). 2 11. For m = 0, 1, 2, 3, ..., am = or 1 (2 − x) cos 1 2 nπx nπx nπx dx + dx + dx x sin (2 − x) sin 2 2 2 −2 −1 1 2 1 nπx nπx dx + dx , (taking advantage of the fact that the (2 − x) sin or bn = x sin 2 2 1 0 product f (x) sin nπx is an even function). 2 For n = 1, 2, 3, ..., bn = 12. u(x, t) = 2e −5π 2 t 16 sin 1 2 −1 (−2 − x) sin πx 2 − e−5π t sin πx 4 2 mπx dx 2 ...
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This note was uploaded on 07/23/2008 for the course MATH 251 taught by Professor Chezhongyuan during the Spring '08 term at Pennsylvania State University, University Park.

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s251final(sp04) - Math 251 Spring 2004 Final Exam ANSW ERS...

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