I have the following parabolic PDE on the domain $[0,T]\times \mathbb{R}$: $$\partial_t u = F(t,x) + (x+\mu(t)) u + (a+bx) \partial_x u +c\partial_x^2 u,$$ where $F$ is a function, $\mu$ only depends on $t$, $a,b$ and $c>0$ are real numbers and the final condition is $u(T,x)=f(x)$. Assume all regularity on $F$, $\mu$ and $f$, if needed.
I have the feeling this equation should be easily solved. I managed to somehow come to an analytic solution applying Fourier transforms which transforms it to a first order PDE and use the method of characteristics. But somehow I believe one should be able to solve it "at once" without Fourier transforms in a much easier way.
I read these notes, but in my case: $B^2 - AC=0$ with $B=0$ so I can not carry out the change of variables suggested in that paper.
Does anyone know how to solve it or what is the smartest way?