## Use Fourier transform to solve PDE damped wave equation

Use Fourier transforms to solve the PDE
$\displaystyle \frac{\partial^2 \phi}{\partial t^2} + \beta \frac{\partial \phi}{\partial t} = c^2 \frac{\partial^2 \phi}{\partial x^2}$, $- \infty < x < \infty$, $t > 0$
subject to
$\phi(x,0) = f(x)$
$\displaystyle \left. \frac{\partial \phi}{\partial t} \right\lvert_{t=0} = g(x)$.

ATTEMPT:

I took Fourier transforms in $x$ and got the ODE
$\Phi_{tt}(k,t) + \beta \Phi_t(k,t) + c^2 k^2 \Phi(k,t) = 0$.

Trying to solve this ODE I get
$\lambda = \frac{-\beta \pm \sqrt{\beta^2 - 4 c^2 k^2}}{2}$.

This results in there are three solutions to the ODE depending on the value of $\lambda$. Am I supposed find solve $\phi$ for all three solutions?

How can I proceed next?

Should I have taken Fourier transforms in $t$ instead in the first step?