When solving PDEs by separation of variables, why are we allowed to divide by the dependent variable? When solving PDEs in physics, one common tool in use is separation of variables. However, to me, there exists a big problem when one divides both sides of a equation by some function that might be $0$.
Here in this link (Laplace's equation 1: Separation of variables, Rudolf Winter, 2008), when we divided by $XY$, should not we assume $XY$ is not $0$? But it seems like $XY$ could be 0 in the general solution. Why is that?
 A: Maybe you can use the separation of variables in $D\backslash U$, where $D$ is the whole domain and $U=\{(x,y)|XY=0\}$. The interior of $U$, denoted by $$\mathring{U}=\{(x,y)\in U| \text{There exists a neighborhood } W \text{ s.t. } (x,y)\in W\subset U\}$$
is empty. Then we can extend the solution to the whole domain by continuity.
Lapalace's equation as an example:
1)  If $X=0, Y\neq 0$, then $Y\frac{{\rm d}^2X}{{\rm d}x^2}+X\frac{{\rm d}^2Y}{{\rm d}y^2}=0$ implies $\frac{{\rm d}^2X}{{\rm d}x^2}=0$, hence the equation
$$
\frac{{\rm d}^2X}{{\rm d}x^2}+k^2X=0
$$ 
still holds.
Similarly for $X\neq0,Y=0$.
2) If $X=Y=0$, I denote $U=\{(x,y)|X(x,y)=Y(x,y)=0\}$, The interior of $U$ is empty(if not , then it is a zero constant solution in an open subset, hence the whole domain $D$). Now you can solve the problem safely in $D\backslash U$ and then extend to the whole domain by continuity.
More general $W=\{(x,y)|XY=0\}$ is a set whose interior is empty, hence we can always safely solve the question in $D\backslash U$ that $XY\neq 0$, then extend to the whole set by continuity. Maybe these thoughts can be modified to explain the validity of separation of variables.
