Show $$\partial_t f(x,t) - a x \partial_x f(x,t) + \frac{1}{2} b \partial_x^2 f(x,t)=0$$ has solutions of the form $$f(x,t) = C(t) e^{-D(t)x^2}$$ and find the differential equations satisfied by $C(t)$ and $D(t)$.
I'm lost on how to begin here. I have not taken a course in PDE's but I would like to know how the above is solved on my way to a larger result in probability.
Is there a "standard" method for such a problem? I've looked at the "ansatz" method which involves taking a guess of the form we wish to show, but I don't see how that would lead to a proof of forms for the solution.