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For the differential equation


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!


marked as duplicate by user61527, ml0105, colormegone, Eric Stucky, M Turgeon Apr 22 '14 at 5:19

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ @GammaFunction yeah I searched for duplicates before posting but didn't find that one. It does look like a dupe, my bad. $\endgroup$ – Platatat Apr 22 '14 at 4:26

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.


Hope this helps a bit.


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