I came across this differential equation $f'(x)=f^{-1}(x)$ on the web, and thought can we do that by a Laplace transform?
But then I got stuck on seeing any clear idea on finding the Laplace transform for $f^{-1}(x)$.
I tried to approach it by differentiating the relation $f^{-1}(f(x))=x$ and then using its result in the $\mathcal{L}\{y(t)\} = \int_0^{\infty}e^{-st}y(t)\,dt$ and then maybe try integration by parts (since I have some form of the derivative of $f^{-1}(x)$ ). But that doesn't seem to work out so well.
So the question is:
If you know the Laplace transform of $f(t)$, $\mathcal{L} \{ f(t)\} = Y(s)$, Can you find the Laplace transform of $f^{-1}(t)$ in terms of $Y(s)$ ?
BTW: The question is not about solving the DE I talked about, I did find out how to solve it.
Best Answer
There is no formula like that. In general, Laplace transforms are useful only for linear differential equations. This one is nonlinear.