Ordinary differential equations

Problem Si. For each n = 0,1,2, … , let fn . —> be the function defined by fn(x) = xn, and let Fn = L[fn(x)] be the Laplace transform of fn(x). Prove by induction on n that
Fn(p) = pn-1-1, p > 0. (Note: for the base case, n = 0, the function is fo(x) = x° = 1 for all x E R. We computed the Laplace transform for this function in lecture.) Hint. In the induction step of your proof, assume that n > 0 and that Fn_i(p) = (n — 1)!/pn. Then, when you compute Fn(p), use integration by parts.

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