Suppose we're trying to solve a functional equation of the form $$Df = f+g,$$ where $D$ means differentiation, $g$ is given, and the goal is to find $f$.
One approach would be to rearrange to $$(D-1)f=g,$$
which becomes
$$f = \frac{1}{D-1} g$$
which becomes
$$f = \frac{-1}{1-D} g$$
which plausibly becomes
$$f = -(1+D+D^2+D^3+\cdots) g.$$
Question. What assumptions do we need on $f$ and $g$ for this line of reasoning to actually work?