A condition for a function to be of separated variables

Question

Let $u(x,y)$ be a $C^2$ function. If $u$ satisfies $u u_{xy}=u_xu_y$, then how can I prove that $u$ is of the form $u(x,y)=f(x)g(y)$? Here the subscript means the partial derivative.

Answer 1

Let $u=e^v$. We have

$$uu_{xy}=u_xu_y\iff e^v(v_{xy}+v_xv_y)e^v=v_xe^vv_ye^v$$ and $v_{xy}=0$. Hence

$$v=f(x)+g(y)$$ and $$u=e^{f(x)}e^{g(y)}.$$