MATH 251 Exam I February 28, 2008 ANSWER KEY 1. 3rd order, linear; 2nd order, nonlinear; 1st order, linear 2. D 3. D 4. C 5. A 6. C 7. D 8. B 9. (a) y = −4, 0, 2 (b) y = −4 is (asymptotically) stable, y = 0 is semistable, and y = 2 is unstable. (c) 0)(lim =∞→tyt (d) 4)(lim −=∞→tyt 10. (a) QtQ8003330+−=′, Q(0) = 0 (b) 301202404)(2+++=ttttP 11. (a) xNyxyM∂∂==∂∂3316 (b) x4y4+ x 2 − 2y = 412. (a) ttcetCeCy5251−−+= (b) Y = (A t + B) e 2t cos 5t + (C t + D) e 2t sin 5t + E e −t (c) Y = (A t 4 + B t 3 + C t 2) e −5t + D t 2 + E t + F 13. (a) u(t) = 2 e −2t cos 4t + e −2t sin 4t (b) µ = 4 (radians/second) (c) True (b/c the system is
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