In $\mathbb R^2$, given $F=F(x,f(x,y))$ and $\partial_{f} g=\frac{1}{x} \partial_{x}(\frac{F}{x})$, in order to have $g=g(f)$, i.e., $g$ is a function of $f$ alone, why do we have to have the following form of $F$? $$F=x^{3}G(f) + xH(f)$$ for some functions $G$ and $H$ of $f$.
We can see $\partial_{f} g=\frac{1}{x} \partial_{x}(\frac{F}{x})=-\frac{F}{x^3}+\frac{\partial_{x}F}{x^2}$ and if we need to have the $F$ in terms of $G$ and $H$, we would have $\partial_{f} g=-G(f) - \frac{H(f)}{x^2}+\frac{\partial_{x}F}{x^2}$. I still couldn't relate them together.