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Let $n$ be a given positive integer and $g$ be a continuous function. We are looking for a function $f \in C^n(\mathbb{R})$ such that $$f^{(n)}-(n+1)f^{(n-1)}-(n+1)nf^{(n-2)}-\dotsc-(n+1)!f=g.$$

It is of course a linear equation of order $n$ but if I try to solve its characteristic equation it gets complicated even for small $n$.

Is there a way to find some operator $L$ (possibly quite "complicated") such that $f = L(g)$?

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I would start looking at solutions for small values of $k$, maybe there's a pattern that allows to formulate a hypothesis or two. –  TZakrevskiy Dec 15 '13 at 0:41

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