= Solution
For the <linearly damped string>, substitute $u(x,t)=X(x)T(t)$ and divide by $XT$:
$$
\frac{X''}{X}=\frac{T''+2\lambda T'}{T}=-\alpha^2.
$$
Thus <separation of variables> gives
$$
\boxed{X''+\alpha^2X=0,\qquad T''+2\lambda T'+\alpha^2T=0}.
$$
The fixed-end <Dirichlet boundary conditions> require $X(0)=X(L)=0$. Nonzero solutions are $X_n(x)=\sin(n\pi x/L)$, with $\alpha_n=n\pi/L$, $n\geq1$. Neither $\alpha=0$ nor a negative spatial <eigenvalue> produces a nonzero function obeying both endpoint conditions.
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