A simple differential equation problem

Question

Given$$\frac{\partial}{\partial u} (Fg)=\frac{\partial}{\partial v} (Fh)$$ where $g,h $ are known functions of $u,v$ and are continuously differentiable. How to guarantee that there exist a function $F(u,v)$ such that the above equation hold?

Answer 1

Use the product rule to obtain : $$Fg_u + gF_u = Fh_v + hF_v \quad \Longleftrightarrow \quad g(u,v)\frac{\partial F}{\partial u} + h(u,v)\frac{\partial F}{\partial v} = -(\frac{\partial g}{\partial u} + \frac{\partial h}{\partial v})F$$

This is can be solved using the method of characteristics. As seen in the link, the equation can be rewritten under the form : $$a(u,v,F)F_u + b(u,v,F)F_v = c(u,v,F)$$ You are guaranteed a solution if the Jacobian $\bigg|\frac{\partial(a,b,c)}{\partial(u,v,F)}\bigg|$ has a non-zero determinant.