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M251ex1(fa03)

# M251ex1(fa03) - Name SID Section Instructor EXAM I MATH 251...

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Name: SID: Section: Instructor: EXAM I MATH 251 October 14, 2003 This is a closed book exam. No notes or calculators may be used. 1 2 3 4 5 6 7 8 9 10 Total: 1

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2 (1) (8 points) For each of the differential equations below, state its order and whether it is linear or nonlinear. (a) y 0 + t 2 y = e t (b) 2 y 00 + 3 y 0 - y = te - t (c) y 0 = y ( y + 1)( y - 1) (d) y 000 - 2 y 0 + ty - y 2 = 0 (2) (5 points) The integrating factor used to solve 2 t 2 y 0 + 6 ty = e - 3 t is (a) e 3 t (b) e 3 t 2 (c) t 3 (d) e - 3 t
3 (3) (5 points) The Existence and Uniqueness Theorem guarantees that the solution to sin( t ) y 00 + 1 t - 3 y 0 + e t y = t 3 , y (1) = 0 , y 0 (1) = 1 is valid on (a) (0 , 3) (b) (0 , π ) (c) ( -∞ , 3) (d) ( -∞ , ) (4) (5 points) Which of the following is the general solution of y 00 + 9 y = 0? (a) c 1 e 3 t + c 2 e - 3 t (b) c 1 e - 3 t + c 2 te - 3 t (c) c 1 e - 9 t + c 2 (d) c 1 cos 3 t + c 2 sin 3 t

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4 (5) (12 points) Solve the initial value problem y 0 = 4 x 3 - 6 x + 3 2 y + 8 , y (1) = - 2 . Give your answer in explicit form.
5 (6) (12 points) For the initial value problem (2 x + ye xy ) dx + ( xe xy + 1) dy = 0 , y (1) = 0 (a) Verify that the equation is exact. (b) Solve the initial value problem. You may leave your answer in implicit form.

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6 (7) (15 points) For the autonomous equation

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