s251ex1(fa04)

s251ex1(fa04) - MATH 251 Fall 2004 Exam I October 12, 2004...

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Unformatted text preview: MATH 251 Fall 2004 Exam I October 12, 2004 ANSWERS: 1. (a) first order, non-linear; (b) second order, non-linear; (c) second order, linear; (d) third order, linear. 2. C 3. C 4. D 5. A 6. C 7. y(t) = −te−t − e−t + 2e−1 . t4 8. (a) The initial value problem describing the process is: Q′ = 3e−t − The solution is Q(t) = 10000 −3 t 300 −t e 100 − e . 97 97 (b) The limiting concentration is 9. (a) 1 Q(t) = lim Q(t) = 0. t→∞ 100 100 t→∞ lim ∂(4xy − 3) ∂(2y 2 + 2x) = 4y = ⇒ it’s an exact equation. ∂x ∂y (b) 2xy 2 − 3y + x2 = C. 10. (a) y(t) = (b) y(t) = −1 −t e sin 2t. 2 t2 + 2 ln t − 1 − 4 ln 2. 4 1 11. α = − . 4 12. (a) y2 = t4 . (b) W (t, t4 ) = 3t4 > 0 through out the interval (0, ∞). 3 Q, 100 Q(0) = 100. ...
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