hw3 - ODE, Homework 3 Due Thursday, June 26, 2008 1....

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ODE, Homework 3 Due Thursday, June 26, 2008 1. (6.3.31) We saw in class that if f is periodic with period T (ie f ( t + T ) = f ( t ) for all t and some fixed minimal positive number T ) then L{ f ( t ) } = R T 0 e - st f ( t ) dt 1 - e - sT . Use this to compute the Laplace transform of the “sawtooth wave” f ( t ) = t for 0 t < 1 and f ( t + 1) = f ( t ) for all t R . 2. (6.4.19 slightly modified) Consider the initial value problem y 00 + y = f ( t ), y (0) = 0, y 0 (0) = 0, where f ( t ) = u 0 ( t ) + 2 n k =1 ( - 1) k u ( t ). (a) Draw the graph of f ( t ) on an the interval 0 t 6 π . (b) Use the Laplace transform to solve the initial value problem. (c) Let n = 3 and plot the graph of the solution for 0 t 6 π . (d) Describe qualitatively what happens to the solutions as n → ∞ . One or two sentences should suffice. 3. (6.5.18 slightly modified) Consider the initial value problem y 00 + y = g ( t ), y (0) = 0, y 0 (0) = 0, where f ( t ) = u 0 ( t ) + 2 n
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