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$$(1) \quad \cfrac{d}{dx} (f(x^n))=\cfrac{-f(x^n)^2}{f(n \cdot x^{n-1})}$$

How do I solve this functional differential equation? I need a closed form solution, so approximations won't cut it, I'll need the whole explicit solution.

I don't have any work, there isn't really an obvious method that pops out at me. However, I do have some context posted here. That's why I need a whole solution.

In general, is there a method to solve certain functional differential equations like $(1)$?

I'm interested in finding the function $f$, so setting $n$ to one or something like that probably wouldn't help. Also I'm particularly interested in the initial condition where $f(0)$ is infinite.

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  • $\begingroup$ Set $n=1$ and it becomes a ordinary differential equation. $\endgroup$ – pregunton Oct 20 '15 at 16:16
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$f(x)=1/x$ is a solution for all $n$. How did I find it? I guessed that there should be a solution of the form $f(x)=x^a$. Are there any others? I do not know.

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  • $\begingroup$ Now that's funny. Given the context, it's a bit embarrassing I didn't start with that guess. Thanks (+1) $\endgroup$ – Zach466920 Oct 20 '15 at 16:25

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