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Unformatted text preview: Differential Equations Test 5 Solutions Answer FOUR questions. Show all working clearly. 1. Find all solutions of d y d x + xy = x. What happens to y as x ? 2. Solve the logistic equation d N d t = rN (1- N/K ) with N (0) = M (the constants r,K,M being positive). 3. Solve the IVP y 00- 2 y + y = 2 e t ; y (0) = 0 ,y (0) = 0 . 4. Solve the mechanical vibration problem m x + kx = F cos t ; x (0) = a, x (0) = 0 given that = p k/m . 5. Calculate the Laplace transforms of the following functions of t : (i) e at , (ii) sin t , (iii) cos t , (iv) t . Solutions : 1. The given equation is linear (and in standard form) with integrating factor exp( Z x d x ) = exp( x 2 / 2) , multplication of the DE by which yields d d x [ y exp( x 2 / 2)] = x exp( x 2 / 2) so that y exp( x 2 / 2) = Z x exp( x 2 / 2)d x = exp( x 2 / 2) + k or y = 1 + k exp(- x 2 / 2) . Of course, this makes it clear that y 1 as x ....
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This note was uploaded on 11/12/2011 for the course MAP 2302 taught by Professor Tuncer during the Fall '08 term at University of Florida.
- Fall '08