Given is the hyperbolic PDE
$$\frac{\partial^2u}{\partial t^2}-c^2 \triangle u=0$$
in the Domain $\Omega\times\Bbb{R}$ using the Dirichlet boundary condition $u=0$ in $\Omega\times\Bbb{R}$. The initial values of this PDE are the position $u(\vec x,t=0)=u_0(\vec x)$ and the velocity $\frac{\partial u}{\partial t}(\vec x,t=0)=u_1(\vec x)$
Show that it is possible to split the PDE into one ODE for the time $t$, and one eigenvalue equation $$-\triangle X(x)=\lambda X(x)$$ for $x \in \Omega$ and $x=0$ in $\partial\Omega$ by using separation of variables.
I know that I have to put all the $t$'s on one side and all the $x$'s (or $\vec x$'s?) on the other side. For the beginning i have $$\frac{\partial^2u}{\partial t^2}=c^2 \triangle u$$
Let $u$ be in the form of $u(x,t)=X(x)T(t)$ $$\frac{\partial^2u}{\partial t^2}=X(x)T''(t)$$ $$X(x)T''(t)=c^2 \cdot\triangle (X(x)T(t))$$ $$\frac{T''(t)}{c^2}=\frac{1}{X(x)}\cdot\triangle (X(x)T(t))$$
This is how far I got. Is this right? How do I handle the $\triangle$ to get the $T(t)$ from the right side to the left?