TITLE: Two functions that are derivatives of each other
QUESTION [3 upvotes]: Is there a pair of functions such that:

$D f(x) = g(x)$
$D g(x) = f(x)$
The functions are not the same function (without this 0 and $e^x$ would work)

In case that there isn't such pair of functions: What is the proof that such pair of functions can't exist?

REPLY [2 votes]: Suppose such functions $f$ and $g$ do exist. Then (letting $D$ be the differential operator) we have $$D^2f(x)=f(x)\\(D^2-1)f(x)=0$$
Since $f$ is not the zero function, then the only way this can happen is if $f$ is a linear combination of eigenfunctions of $D^2-1$ with eigenvalue $\pm 1$. (Why?) That is, $f(x)=ae^{x}+be^{-x}$ for some constants $a,b$. From this you can deduce $g(x).$
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