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s251ex1(fa00) - t − 3 t cos t 3 sin t − 3 = − 3 t cot...

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MATH 251 Fall 2000 Exam I September 26, 2000 ANSWERS: 1. (a) F = a 0 ( t ) y + a 1 ( t ) y + a 2 ( t ) y ′′ + ... + a n ( t ) y ( n ) + g ( t ) , a n ( t ) negationslash = 0 . (b) (i) second order and linear; (ii) third order and non-linear; (iii) first order and linear. 2. (a) y ( t ) = αe t αe 2 t . (b) All real numbers; all values of α would make the limit 0. (c) α = 0 only. 3. (a) y = 0 and y = 3 . (b) y = 0 is a stable equilibrium solution, y = 3 is an unstable equilibrium solution. (c) Since y = 0 is an equilibrium solution, it’s a constant solution. Therefore, if y (0) = 0 then y (1) = 0 . 4. (a) y ( t
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Unformatted text preview: t [ − 3 t cos t + 3 sin t − 3] = − 3 t cot t + 3 − 3 csc t . (b) (0 , π ) is the largest interval. 5. (a) y 2 = x 3 3 + C . (b) x 2 2 − xy + y 2 2 = C , or (a±ter frst dividing both sides by x − y ), y = x + C . 6. (a) Q ′ = − 20 1100 Q , or Q ′ + 1 55 Q = 0 ; Q (0) = 100 where Q ( t ) = the amount o± blue M&Ms in the vat at time t . (b) Q ( t ) = 100 e − t 55 . (c) t = 55 ln( 100 11 ) ....
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