Use the method of separation of variables to find the natural frequencies (i.e the $\displaystyle \Omega$ in the equation $\displaystyle T''+\Omega^2T=0$ where T=T(t) ) for the waves which satisfy the equation
$\displaystyle U_{tt}-c^2U_{xx}+kU=0, 0\leq x \leq l,$
where k>0 and c>0 are constants with
$\displaystyle U(0,t)=U(l,t)=0.$

I can get to the point of separating the equation to get 2 DE'S to solve, but I'm a bit stuck as to where to go from there. I've got that,
$\displaystyle X''-\alpha X-\frac{k}{c^2}=0$
$\displaystyle T''-c^2 \alpha T=0$

Using the boundary conditions I get that X(0)=0 and X(l)=0, should I use these to solve the first equation involving X?

Any help much appreciated!