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s251final(fa03) - MATH 251 Fall 2003 Final Exam ANSWERS 1 B...

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MATH 251 Fall 2003 Final Exam Decemebr 15, 2003 ANSWERS: 1. B 2. B or C 3. C 4. A 5. B 6. A 7. (a) Critical points are (1 , 1) and (1 , - 1) (b) (1 , 1) is unstable (a saddle point); (1 , - 1) is unstable (a spiral point/spiral source) 8. y ( t ) = - 7 6 e - 3 t + 1 2 e t - 1 3 t + 2 3 9. m = 2 , γ = 6 , k = mg L = 4 (a) 2 u ” + 6 u 0 + 4 u = 0 or u ” + 3 u 0 + 2 u = 0 , u (0) = 0 , u 0 (0) = 5 Solution: u ( t ) = 5 e - t - 5 e - 2 t (b) (Trick question!) The system is overdamped . There is no oscillation, therefore, there is no quasi- period. Also acceptable: quasi-period = . 10. (a) y = - 3 , 0 , 3 (b) y = - 3 is asymptotically stable; y = 0
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