# Question:Duhamel's Convolution Principle

## Question:Duhamel's Convolution Principle

I've been trying to find information about this apparently elusive (or perhaps obscure) topic without a lot of luck. I was curious if anyone had ever heard of a technique for solving an nth-order nonhomogeneous linear ODE with constant coefficients by means of Duhamel's Convolution Principle. The guy who taught this technique to me didn't believe in using textbooks (or staying in accordance with predefined curricula) and he basically just taught whatever he wanted. He essentially claimed that any nth-order nonhomogeneous linear ODE with constant coefficients could be solved by first solving the associated homogeneous equation, and then, writing a new function (he usually called it theta(t)) using the same form as the solution to the associated homogeneous equation but with initial conditions such that, for t = 0, theta of t, theta prime of t, theta double prime of t, etc., all equaled 0, except for the final condition (theta^(n-1)(t)) which equaled 1. So, you construct this function, theta(t), using the aforementioned initial conditions, and the solution to your ODE is the sum of the solution to the associated homogeneous equation and the convolution of theta(t) and your forcing function (i.e., the integral from 0 to t of the product of theta(t-tau) and f(tau) with respect to tau, where f is your forcing function). Although I have found bits and pieces of information, I have never found any documentation of the validity of this method; the only information I have found suggests that Duhamel's Convolution Principle is used for inverting Laplace transforms, but, I have not found any specifics. Any information anyone has would be greatly appreciated. Thanks.
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