Use Fourier transforms to solve the PDE

, ,

subject to

.

ATTEMPT:

I took Fourier transforms in and got the ODE

.

Trying to solve this ODE I get

.

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

How can I proceed next?

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