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[tex]\frac{\partial x}{\partial t} = A \frac{ \partial x}{\partial y} + B \frac{\partial^2 x}{\partial y^2}[/tex]

I know how to solve the diffusion equation (i.e. no dx/dy term), but that method doesn't work here. I tried to go with the LaPlace Transform route, but I got an ugly term of the following form:

[tex]\frac{-A-\sqrt{4 B s+A^2}}{2B} y [/tex]

Which I can't find a handy inverse LaPlace for, and which Mathematica doesn't give a real answer to.

Any suggestions how to approach this? Thanks.