Problem statement:
We have a Brownian particle in harmonic potential with additional time-dependent force. Langevin equation(mass taken to be unit):
$\ddot{x} + \gamma \dot{x} + \omega_0^2 x = \epsilon \cos(\omega t) + \sqrt{2D} \xi(t)$
Corresponding Fokker-Planck equation for $P = P(x,v,t)$, with $v = \dot{x}$:
$\partial_t P = - v\partial_x P + \partial_v(\gamma v + \omega_0^2 x - \epsilon \cos(\omega t)) + D\partial_v^2 P$
My attempt at solving:
Perform bilateral Laplace transform on $x$ and $v$. This would lead to a differential equation only involving derivatives in $t$. We would solve that and then preform inverse Laplace transform to get back dependency on $x$ and $v$.
The problem is that Laplace transform doesn't get rid entirely of derivatives in $v$ and $x$ since for example
$v\partial_x P \Rightarrow \hat{x}\partial_{\hat{v}}\hat{P}$
Because of that, we would first have to make some sort of variable substitution:
$y = y(x,v)$
$z = z(x,v)$
and possibly
$F = u(x,v) P (x,v,t)$
which would than lead to PDE without variables as prefactors, sth like:
$\partial_t F = c_1 \partial_x F + c_2(t) \partial_v F+ c_3 F + c_4 \partial_v^2 F$
and then continue as described.