# Use Fourier transform to solve PDE damped wave equation

#### math2011

This question is also posted at http://www.mathhelpforum.com/math-help/f59/use-fourier-transform-solve-pde-damped-wave-equation-188173.html

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?

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