Solve one dimensional wave equation using fourier transform I'm trying solve this wave equation using Fourier method, but I am stuck...
$${ u }_{ tt } ={ c }^{ 2 }{ u }_{ xx } - \alpha{ u } =0, \  0<x\le L, t  >0 $$
$${ u }( 0,t) = { u }( L,t) = 0$$
$${ u }( x,0) = f(x), { u }_{ t }( x,0) = g(x) $$
I know that first I have to use separation of variables:
$${ u }( x,t) = T(t)X(x). $$
Making the calculations
$$\frac{T''+ \alpha T}{c^{2}T} = \frac{X''}{X} = -\lambda  $$
I guess I'm right at this point? Okay? Now I have to solve:
$$X'' + \lambda X = 0,$$
and
$$\frac{T'' + \alpha T}{c^{2}T}  = -\lambda ,$$
$$ T'' + (\alpha + \lambda c^2)T = 0.$$
I don't now how to solve the second equation and how I add the two equation to solve the first problem. 
I will be very grateful for the help!!!!
 A: When you solve the equation for $X$ for $\lambda =\alpha^2>0$, this gives you 
$$
X=A\sin(\sqrt{\alpha}x)+B\cos(\sqrt{\alpha}x).
$$
Applying the initial conditions gives
$$
0=A\sin(\sqrt{\alpha}L)
$$
which implies that either $A=0$ or $\sin(\sqrt{\alpha}L)=0$. Note that $A=0$ gives a trivial result so we must have $\sin(\sqrt{\alpha}L)=0,$ ie, $\alpha_nL=n\pi,~n=1,2,3,\cdots$. Corresponding to each $n\in\mathbb{Z}^+,~\exists$ eigenvalue $\lambda_n=\alpha^2=\frac{n^2\pi^2}{L^2},~n=1,2,3,\cdots$. This implies that the solution is 
$$
X_n(x)=A\sin\left(\frac{n\pi x}{L}\right)
$$
were $\lambda_n$ are eigenvalues and the function $B_n\sin(\frac{n\pi x}{L})$ is the corresponding eigenfunction.
With this information, we can go after the equation for $T.$ For given $\lambda_n,$ we have
$$
T''_n+\left( \alpha+\frac{n^2\pi^2c^2}{L^2} \right)T=0
$$
which has Auxiliary equation
$$
r^2+\alpha+\frac{n^2\pi^2c^2}{L^2}=0 \\
\Rightarrow r^2=\pm i\sqrt{\alpha+\frac{n^2\pi^2c^2}{L^2}}.
$$
Solving this ODE gives
$$
T_n(t)=a_n\cos\left( \sqrt{\alpha+\frac{n^2\pi^2c^2}{L^2}}t \right)+b_n\sin\left( \sqrt{\alpha+\frac{n^2\pi^2c^2}{L^2}}t \right),~n=1,2,3,\cdots
$$
After applying the final initial conditions, we can write the solution
$$
u_n(x,t)=X_n(x)T_n(t)
$$ 
and take ay linear combination of the functions $u_n$. Is that what you were looking for?
