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Use Fourier transforms to solve the PDE

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

subject to

[tex]\phi(x,0) = f(x)[/tex]

[tex]\displaystyle \left. \frac{\partial \phi}{\partial t} \right\lvert_{t=0} = g(x)[/tex].

ATTEMPT:

I took Fourier transforms in [tex]x[/tex] and got the ODE

[tex]\Phi_{tt}(k,t) + \beta \Phi_t(k,t) + c^2 k^2 \Phi(k,t) = 0[/tex].

Trying to solve this ODE I get

[tex]\lambda = \frac{-\beta \pm \sqrt{\beta^2 - 4 c^2 k^2}}{2}[/tex].

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

How can I proceed next?

Should I have taken Fourier transforms in [tex]t[/tex] instead in the first step?