For the differential equation
$$\frac{d}{dx}[y(x)]=y^{(-1)}(x)$$
where $y^{(-1)}(x)$ is the inverse of $y(x)$, find y(x).
I gave up on finding the solution analytically pretty quickly and decided that a numerical approach might be more effective. But, I'm not sure that this problem can be solved, even numerically; an Euler or Runge-Kutta type method will not work because to find the value of $y^{(-1)}(a)$, one must first know the value of $y(b)$, where $a$ is not necessarily equal to $b$. Sort of like trying to solve $\frac{d}{dx}[y(x)]=y(x+1)$, I don't know of any numerical approaches that can handle a problem of that type.
If anyone has any ideas on how this might be solved (or proven unsolvable), they would be appreciated. Thanks!
You are not the first to have pondered such a question. The following may be of interest to you:
i) For bijective $f:x \rightarrow y(x)$. Elementary maths, easy to understand:
Inverse of a bijection f is equal to its derivative
ii) For $f$ differentiable on $(0,\infty)$. This answer, being on overflow, is somewhat beyond my ken anywho. The solution involving the golden ratio, however, is fine.
https://mathoverflow.net/questions/34052/function-satisfying-f-1-f/34095#34095
Hope this helps a bit.