Handle a function of itself I have a tricky question. I want to handle a function y(x) defined in this way:
$y(x)=f(y(x))g(x)$
Here, $f(y(x))$ and $g(x)$ are smooth functions: is there any way/method to express $y(x)$ as a (complicated) function of $x$ solely?
Maybe it is impossible if $f$ and $g$ are generic.
Edit:
If I have the numerical functions which describe $f(y)$ and $g(x)$, is there any approximated way to express $y(x)$? For example with recursive formulae?
My point is not to guess what form $y(x)$ should have, but deriving it or, at least, inferring it approximately.

You can find many functions to satify your conditions.
For example,on $x\in[0,\infty)$ define  $$y(x)=x^2,f(x)=\sqrt x, g(x)=x$$
We get $$f(y(x))g(x) =x\sqrt {y(x)} = x^2= y(x)$$
You may design other examples with little effort.    
