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s251final(sp04)

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

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Math 251 Spring 2004 Final Exam ANSWERS 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 2 t + Bt 2 + Ct + D . 3. a) T = 2 π 3 (sec); b) μ = 5 (rad/sec); c) It is critically damped. 4. y ( t ) = u 1 ( t )( t - 1) e 3( t 1) - 2 u 3 ( t )( t - 3) e 3( t 3) 5. braceleftbigg x 1 = x 2 x 2 = 5 2 x 1 + x 2 or x = bracketleftbigg 0 1 5 2 1 bracketrightbigg x 6. a) x ( t ) = C 1 bracketleftbigg 1 2 bracketrightbigg e 4 t + C 2 bracketleftbigg 3 1 bracketrightbigg e t b) It is an unstable saddle point. c) α = - 1 d) 7. a) x ( t ) = bracketleftbigg cos t - 4sin t 4cos t + sin t bracketrightbigg e 3 t b) It is an unstable spiral point. 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. braceleftbigg λX ′′ - X + 5 x 3 X = 0 λT - T = 0 10. The eigenvalues are λ = 1 4 . 4 4 , 9 4 , ..., n 2 4 , ... Their eigenfunctions are y n ( x ) = C n cos nx 2 , n = 1 , 2 , 3 , 4 , ... b) Yes, λ = 0 is an eigenvalue. Any nonzero constant function is a corresponding eigenfunc- tion. 1

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11. For m = 0 ,
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s251final(sp04) - Math 251 Spring 2004 ANSW ERS Final Exam...

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