Consider the equation $$ \left(\frac{\partial f}{\partial r}\right)^{2} + \frac{1}{r^{2}}\left(\frac{\partial f}{\partial \varphi}\right)^{2} = g(r). $$ How to solve it? Here the left hand side is $|\nabla f|^2$ in polar coordinates.
After separating the variables, I get $$\sqrt{g(r)r^{2} - r^{2} (\partial_{r} f)^{2} } = \partial_{\varphi} f$$