I am trying to solve the following PDE that comes from a Fokker-Planck equation: $$\partial_t g(x,t) = x^2 f_2(t) \partial_{xx} g(x,t) + x f_1(t) \partial_x g(x,t) + f_0(t) g(x,t), $$ where $f_0,f_1,f_2$ are known smooth functions of $t$.
The stationary version of this equation, in which $f_0, f_1, f_2$ are constant and $\partial_t g(x,t) \equiv 0$, I can solve by guessing and checking a solution proportional to $x\mapsto x^\zeta$ for some power $\zeta$.
However is there an analytic solution for the nonstationary case? What about in the special case where $f_0,f_1,f_2$ are constant but we look for a solution with $\partial_t g(x,t) \not\equiv 0$ ?
I'm sure equations like these have been studied but I'm not really sure where to start looking. Thanks for the help.