Many posts on $SE$ are related to solving Laplace's equation and I guess I didn't find the right one to answer my question; so, excuse me if it seems obvious, but there is something I don't understand...
It' related to this document: here, in which (first page) the author says:
" Recall that Laplace's equation in cylindrical coordinates is given by $$\frac{\partial^2u}{\partial\rho^2}+\frac{1}{\rho}\frac{\partial u}{\partial\rho}+\frac{1}{\rho^2}\frac{\partial^2u}{\partial\phi^2}+\frac{\partial^2u}{\partial z^2}=0$$ while substituting in the separated $u=P(\rho)\Phi(\phi)Z(z)$ and dividing by $u$ gives the equation $$\frac{P^"}{P}+\frac{P^{'}}{\rho P}+\frac{1}{\rho^2}\frac{\Phi^"}{\Phi}+\frac{Z^"}{Z}=0$$ from which we see that we must have both $\frac{\Phi^"}{\Phi}$ and $\frac{Z^"}{Z}$ constant "
I just don't understand why the last sentence "from which we see that we must have both $\frac{\Phi^"}{\Phi}$ and $\frac{Z^"}{Z}$ constant" seems to be so obvious to the author.
Could someone elaborate a bit on this and explain more clearly why "we must have"? I guess I'm stuck on something elementary about differential equations, but that's what precisely escapes me...