What Are All Separable Solutions of the Laplace Equation? Let us call a separable solution of the Laplace equation $\frac{\partial^2\Phi}{\partial x^2}(x,y)+\frac{\partial^2\Phi}{\partial y^2}(x,y)=0$ as a solution of the form $\Phi(x,y)=X(x)Y(y)$ which satisfies this equation. My question is that what are all such solutions?
I considered the following theorem which states a necessary condition for such solutions.

Theorem. Suppose that $\Phi:[-a,a]\times[-b,b]\to\mathbb{R}$, $X:[-a,a]\to\mathbb{R}$, $Y:[-b,b]\to\mathbb{R}$ are functions such that $\Phi(x,y)=X(x)Y(y).$ Assume that $X\in C^2[-a,a]$ and $Y\in C^2[-b,b]$ and none of them is the zero function. The following is valid
  \begin{align*}
\frac{\partial^2\Phi}{\partial x^2}(x,y)+\frac{\partial^2\Phi}{\partial y^2}(x,y)=0\implies
\begin{cases}
\frac{d^2X}{dx^2}(x)+\lambda X(x)=0 \\
\frac{d^2Y}{dy^2}(y)-\lambda Y(y)=0
\end{cases}
\end{align*}
  for every $(x,y)$ belonging to $[-a,a]\times[-b,b]$. Also, $\lambda$ is some arbitrary real number.

Proof. Noting that $\frac{\partial^2\Phi}{\partial x^2}(x,y)=\frac{d^2X}{dx^2}(x)Y(y)$ and $\frac{\partial^2\Phi}{\partial y^2}(x,y)=X(x)\frac{d^2Y}{dy^2}(y)$ and putting it into the Laplace equation we arrive at
\begin{align*}
\frac{d^2X}{dx^2}(x)Y(y)+X(x)\frac{d^2Y}{dy^2}(y)=0,\tag{1}
\end{align*}
which holds for every $x$ and $y$ belonging to $[-a,a]\times[-b,b]$. Dividing both sides by $X(x)Y(y)$, we conclude that
\begin{align*}
\frac{\frac{d^2X}{dx^2}}{X}(x)+\frac{\frac{d^2Y}{dy^2}}{Y}(y)=0
\end{align*}
should hold for those $(x,y)$ belonging to $[-a,a]\times[-b,b]$ such that $X(x)Y(y)\ne 0$. This leaves no other choice but
\begin{align*}
\begin{cases}
\frac{\frac{d^2X}{dx^2}}{X}(x)=-\lambda \\
\frac{\frac{d^2Y}{dy^2}}{Y}(y)=\lambda.
\end{cases}
\end{align*}
for the aforementioned $x$ and $y$. Multiplying by $X(x)Y(y)$ and rearranging the terms give the desired result.
Questions
$1$. I was thinking what if $X(x)Y(y)=0$ for some $(x,y)$? For example, take $\lambda=-\omega^2$ and solve the ODEs, then some function like $\Phi(x,y)=\sinh(w x)\sin(w y)$ satisfies the Laplace equation and the ODEs. It also vanishes whenever $x=0$ or $y=0$. However, my proof seems not to be working at such points.
A natural question to be asked is 

Do all of the separable solutions $\Phi(x,y)=X(x)Y(y)$ of the Laplace equation, should satisfy that pair of ODEs?

I know that the answer is YES, but I don't know how to make an argument about that. Or maybe I am making a stupid mistake around here!?
$2$. Is there any other type of argument which may avoid these things?
Recently, I found that I have asked a similar question, which is slightly different, about an year ago but I did not receive a good answer.
Any hint or help is appreciated. :)

When you separate variables, you are only constructing some special solutions, you are not trying to get all solutions. That's why you do not worry too much when assuming that no function vanishes too often.
A posteriori, after you have constructed enough solutions with separate variables, you will see if you can obtain the general (and unique) solution to your original boundary value problem as a superposition of the special solutions you found in the previous step. In the case at hand, it turns out that the special solutions you constructed before suffice (indeed, this is the rediscovery of Fourier series). Therefore, there is no need to worry about the non-vanishing assumptions you made in the previous step. 
