# How to Solve dx/dt = Adx/dy + Bdx^2/dy^2

Hello, I am trying to solve the following equation:

$$\frac{\partial x}{\partial t} = A \frac{ \partial x}{\partial y} + B \frac{\partial^2 x}{\partial y^2}$$

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:

$$\frac{-A-\sqrt{4 B s+A^2}}{2B} y$$

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.

$$x(y,t)=F(y)*G(t)$$