Solving Kramers Equation PDE - Find a Solution

[WarnK]

Homework Statement

Homework Equations

Find a solution to the PDE
$$B P_{vv} - v P_x + (A v - F(x)) P_v + A P = 0$$
where A and B are constants, P = P(x,v)

The Attempt at a Solution

I have no idea how to even guess a solution to this.

 

[Kreizhn]

I would suggest separation of variables, but that F(x) seems to screw things up a bit...

 

[WarnK]

Assumeing $$F(x) = -V_x$$ and makeing an ansatz
$$P(x,v) = C_1 exp(C_2 x + C_3 V(x) + C_4 v + C_5 v^2)$$
I get these conditions on the constants C_i
$$(2BC_5+A)C_5=0$$
$$(4BC_5+A)C_4-C_2+(C_3+2C_5)F(x)=0$$
$$(BC_4-F(x))C_4+2BC_5=0$$
in the second eq we can put C_3=-2C_5 and get rid of F(x) there, but it's still in the third eq. Suggestions?