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Homework Assignment 16 in Differential Equations, MATH308-Fall 2015 This homework is not for the submission but it is highly recommended to solve it because the material of it is in the final test Topics covered : step function,Laplace transform of discontinuous functions and differential equations with discontinuous forcing function(corresponds to sections 6.3, 6.4), convolution interal (section 6.6) 1. Find the Laplace transform of the function t < 2, 5t 2 f (t) = t + t − 2 2 ≤ t < 6, 1 + 4t 6 ≤ t. π 2. Find the inverse Laplace transform of the function e− 2 s (3s2 − 2s + 1) . (s + 1)2 (2s2 + 12s + 26) 3. Use the convolution theorem to find the inverse Laplace transform of the given function: s2 (s2 + 9)(s2 + 25) 4. Find the solution of the initial value problem y 00 + 8y 0 + 25y = g(t); y(0) = 1, y 0 (0) = −2, where g(t) = 2 cos 2t, 3π 2 3π t≥ . 2 0≤t< 1 − 3 sin 2t, 5. (a) Express the solution of the given initial value problem in terms of a convolution integral: y 00 − 10y 0 + 29y = g(t), y(0) = 0, y 0 (0) = −2. (1) (b) Find the solution of the same initial value problem (1) using the method of variation of parameter. Show that your answer coincides with the answer obtained in item (a). 1