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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!

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  • $\begingroup$ @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
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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.

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