s251final(fa00)

s251final(fa00) - Answers to Fall 2000 Final exam. Q#:1...

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Answers to Fall 2000 Final exam. Q#:1 y ( t )= e 2 t ± t 3 3 - 1 ² Q#:2 A y ( t c 1 e - 2 t cos t + c 2 e - 2 t sin t B y ( t c 1 e 3 t + c 2 te 3 t C y ( t c 1 e 4 t + c 2 e - 5 t Q#:3 y p ( t At cos3 t + Bt sin 3 t + Cte t + De t + Et 2 + Ft + G Q#:4 y ( t 3 2 sin 2 t + 1 2 u π ( t )sin(2 t - 2 π 3 2 sin 2 t + 1 2 u π ( t )sin2 t Q#:5 A X ( t c 1 ³ - 4 1 ´ e - 4 t + c 2 ³ 1 2 ´ e 5 t B Saddle point, which is unstable. C D X ( t 1 9 ³ - 4 1 ´ e - 4 t - 5 9 ³ 1 2 ´ e 5 t Q#:6 A The critical points are P =(2 , 2) and Q =( - 2 , 2). B Point P: X 0 = ³ 01 4 - 4 ´ X . λ 1 = - 2+2 2and λ 2 = - 2 - 2 2. Point Q: X 0 = ³ - 4 - 4 ´ X . λ = - 2. B Point P is a Saddle point, which is unstable. Point Q is an improper stable node, which is stable. Q#:7 u ( x, t )=exp[ - 9 π 2 t 25 ]sin= frac 2 πx 5 - 2exp[ - 9 π 2 t ]sin +13exp[ - 19(49) π 2 t 25 7 5 Q#:8
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