$$\frac{1-f\Big(\frac{x}{x+(1-x)f(x)}\Big)}{1-f(x)} = 1-x(1-x)\frac{f'(x)}{f(x)}$$
Now, we know that $f(x)=c$ and $f(x) = \frac{a+bx}{1-x}$ are two solutions. How can I get other solutions or to prove that these are all the solutions? Thanks!
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I would substitute $$t=\frac{x}{x+(1-x)f(x)}$$ – Dr. Sonnhard Graubner Mar 27 '19 at 19:02
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This may be a bit late but technically these are not all solutions per se: the constants are not all solutions because for most of them the denominator of $\frac{x}{x + (1 - x)c}$ vanishes for some $x \neq 0$, and the second family aren't solutions because their denominator vanishes at $x = 1$. Granted, these could be seen as mere technicalities, especially the second one since you could just restrict the domain (the domain and codomain should always be indicated for these kinds of reasons!), but these technicalities need to be taken into account. – Bruno B Dec 14 '23 at 09:08