Proof to the general solution of a PDE

Question

Assume that $f(u,v)$ is a two-variable function. How do I solve this PDE? $$\frac{1}{f}\frac{\partial{f}}{\partial v}=\frac{-2}{v}$$ What I did was pretending that the partial derivative is a normal derivative, in this case $f=Cv^{-2}$ Then I notice that I can multiply the entire thing by an arbitary function $g(u)$, so in the end my guess is $f(u,v)=g(u)v^{-2}$. This seems to work, but how to I prove that it's the MOST GENERAL solution?