s251ex1(sp07) - (b Yes Because W y 1,y 2 t = te t − e t n...

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MATH 251 Spring 2007 Exam I March 1, 2007 ANSWERS: 1. A 2. E 3. ( 1 , 1) 4. (a) y = 4 , 0 , 2 ; (b) y = 0 is (asymptotically) stable; (c) lim t →∞ y ( t ) = 0 . 5. (a) ∂M ∂y = 1 = ∂N ∂x ; (b) x 4 4 + xy + 5 y 2 2 = 10 . 6. (a) y ( t ) = C 1 e 2 t + C 2 e t + 3 te t ; (b) y ( t ) = 2 3 e 2 t + 2 3 e t + 3 te t . 7. (a) sin y + y 2 2 = e t + t + sin3 + 7 2 ; (b) y = 3ln t + 2 t or y = ln t 3 + 2 t . 8. y ( t ) = C 1 e t cos 2 t + C 2 e t sin 2 t . 9. (a) Substitute y 1 and y 2 into the equation to see that they would each satisfy it.
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Unformatted text preview: (b) Yes. Because W ( y 1 ,y 2 )( t ) = te t − e t n = 0 . (c) y ( t ) = C 1 t + C 2 e t . 10. W ( y 1 ,y 2 )( t ) = 4 t 2 11. (a) Q ′ = 10 − 1 50 Q , or Q ′ + 1 50 Q = 10 , Q (0) = 0 ....
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