Finding Solutions to $\frac{\partial^2 V}{\partial u \partial v} = 0$

[PhDorBust]

Given $$\frac{\partial^2 V}{\partial u \partial v} = 0$$, the solution is $$V_1(u) + V_2(v)$$ for arbitrary $$V_1 , V_2$$.

I solve to get $$\frac{\partial V}{\partial u} = V_3(u)$$ and then $$V = \int V_3(u) du + C$$ Where V_3, C are arbitrary.

How could I transform my latter solution into the first solution? Don't V_1, V_2 have to have some properties such as differentiability? (I found this in a physics textbook)

 

[lanedance]

try the same method but integrate over the other variable first then equate the two results