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?