When I was playing around with functional equations, I came up with a problem which has a simple trivial answer.
I'm talking about the following equation:
$f(x)+f^{-1}(x)=f'(x)+\ln x$
The trivial solution is $f(x)=e^x$, but it's not clear to me how would one go about solving for $f(x)$ from scratch.
Is it even possible to solve such an equation without guessing? Are there any methods that can be used to tackle functional differential equations involving the inverse function?