Fourier transform of a differential equation

[AxiomOfChoice]

Homework Statement

I'm supposed to take the "spatial Fourier transform" of the partial differential equation

$$p_t = \frac{a^2}{2\tau}p_{xx} + 2g(p + xp_x)$$

for $p = p(x,t)$.

Homework Equations

Well, I guess I eventually need something like

$$\phi(k,t) = \mathbb F(p(x,t)) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} p(x,t) e^{-ikx}\ dx.$$

The Attempt at a Solution

I've never done anything like this before, so I'm really not sure what to do. I think if I get a few hints or a gentle nudge in the right direction, I'll be good to go, though.

I'm guessing that I should take the spatial Fourier transform $\mathbb F$ of both sides. So the LHS would become

$$\mathbb F(p_t) = \frac{1}{\sqrt{2 \pi}} \int \frac{\partial p}{\partial t}e^{-ikx}dx = \frac{1}{\sqrt{2 \pi}} \frac{\partial}{\partial t} \int p(x,t) e^{-ikx}dx = \phi_t.$$

Is this right?

 

[AxiomOfChoice]

And then I guess we can use the rule that

$$\mathbb F(p_{xx}) = -k^2 \mathbb F(p)$$

to conclude that the transform of the first term on the RHS is just

$$-\frac{a^2k^2}{2\tau} \phi.$$

So, at the end of the day, our differential equation becomes

$$\phi_t = -\frac{a^2k^2}{2\tau} \phi + 2g(\phi + \mathbb F(xp_x)).$$

That last term on the right is giving me fits!