There is no exact answer for this question, but I'm having trouble with guessing what the author intended.
Solve the boundary-value problem:
$u_{tt} = c^2 u_{xx} + u$
Boundary condition: $u(x,0)=f(x), u_t(x,0)=0, u(0,t)=0, u(l,t)=0$.
This problem is an exercise in an elementary differential equation text and I don't get what the author intended.
Here's how I tried this:
Set $u(x,t)=X(x)T(t)$.
Then, $\frac{X''}{X}=\frac{T'' - T}{c^2 T}$.
Thus for some constant $\lambda$, $X''+\lambda X =0 $ and $T'' + (c^2\lambda - 1)T=0$.
So $\lambda=\lambda_n=\frac{n^2\pi^2}{l^2}$ and $X(x)=X_n(x)=\sin (\frac{n\pi x}{l})$.
However, I have a trouble with finding $T_n$. There may be a case that $c^2 \lambda_n - 1 \leq 0$ for some $n$. So I think it should be assumed that $c^2 > l^2/\pi^2$.
If this is not the author intended, how do I solve this?