How would you go about solving the damped equation:
$ u_{tt} = u_{xx} - u_t $
$0 < x < 1,$ $t > 0$
With intial conditions: $ u(x,0) = \sin(2 \pi x)$ and $u_t(x,0) = 0$
With boundary conditions: $u(0,t) = 0$ and $ u(1,t) = 0$
So far, I am able to separate $u(x,t) = \phi(x)G(t)$, and solve: $ \frac{G'' + G'}{G} = \frac{\phi''}{\phi} = -\lambda $ to get $\lambda_n = (n\pi)^2$ and $\phi_n = \sin(n \pi x)$ but run into a dead end afterward.
Since the Fourier series for $u(x,0)$ has only the one term ($n=2$), we may take $\phi(x) = \sin(2\pi x)$, $\lambda = 4 \pi^2$. The solutions for $G(t)$ can be written as $e^{-t/2} (c_1 \cos(\alpha t) + c_2 \sin(\alpha t)) $ where $\alpha = \sqrt{4\pi^2-1/4}$. Then determine $c_1$ and $c_2$ so the initial conditions are satisfied.