For my research, I would like to solve the equation $$\frac{\partial^{2} f}{\partial x \partial y}= u\big(f(x,y)\big)\frac{\partial f}{\partial x}\frac{\partial f}{\partial y},$$ where $u$ is a generic function of $f(x,y)$.
Please, does anyone know if an explicit representation of the solution $f(x,y)$ exists? In general, I'm interested in characterizing $f(x,y)$.