Question

For full credit, you must show all work and box answers 1. If functions f and g are piecewise continuous on the interval [0, oo), then the convolution of f and g is a function defined by the integral The Convolution Theorem (theorem 7.4.2 in your book and formula 6 in your table) states: If j(t) and g) are piecewise continuous on [0, oo) and of exponential order, then We are going to use convolution to solve y"-y,-t-e-,, y(0)-0, y'(0)-0. (a) Solve g" -g'-(, 9(0)-, (. This solution is called the impulse response of the system (b) The solution to y" y,一f(t), y(0)-0, y'(0) = 0 is the convolution of the external force f(t) and the impulse response of the system g(t) from part (a). Show this by solving the initial-value problem with the non-specified (t) using Laplace transforms and the convolution theorem. Your answer will have f(t) in it (c) Solve y" y' = t-e t y(0) = 0, y'(0)-0 using Laplace transforms and the convolution theorem. You can use your result from part (b), but evaluate the convolution integral now that you know f(t). For more information on convolution, check out the "Interesting Websites" page in Canvas

Answer 1

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