Use Fourier transforms to solve the PDE

$\displaystyle \displaystyle \frac{\partial^2 \phi}{\partial t^2} + \beta \frac{\partial \phi}{\partial t} = c^2 \frac{\partial^2 \phi}{\partial x^2}$, $\displaystyle - \infty < x < \infty$, $\displaystyle t > 0$

subject to

$\displaystyle \phi(x,0) = f(x)$

$\displaystyle \displaystyle \left. \frac{\partial \phi}{\partial t} \right\lvert_{t=0} = g(x)$.

ATTEMPT:

I took Fourier transforms in $\displaystyle x$ and got the ODE

$\displaystyle \Phi_{tt}(k,t) + \beta \Phi_t(k,t) + c^2 k^2 \Phi(k,t) = 0$.

Trying to solve this ODE I get

$\displaystyle \lambda = \frac{-\beta \pm \sqrt{\beta^2 - 4 c^2 k^2}}{2}$.

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

How can I proceed next?

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