What trick is needed to solve this second order, linear, two dimensional differential equation? How can I solve this differential equation?
$$
f_{xx}(x, y) + f_{yy}(x, y) - xf_x(x, y) - yf_y(x, y) - f(x, y) = 0
$$
If it were instead 
$$
f_{xx}(x, y) + f_{yy}(x, y) - xf_x(x, y) - yf_y(x, y) - 2f(x, y) = 0
$$
Then it would be easy to give solutions in terms of the solutions to the ODE
$$
g''(y) - yg'(y) - g(y) = 0.
$$
But I am missing a term of $2f$ -- is there a "trick" that I can use to achieve it, given that I know how to solve a very related differential equation?

Separation of variables:
$f(x,y) = X(x)Y(y)$ is a solution where for some constant $c$, 
$X''(x) - x X'(x) + c X(x) = 0$ and $Y''(y) - y Y'(y) - (1 + c) Y(y) = 0$.
The general solution to $X''(x) - x X'(x) + c X(x) = 0$ is 
$$X(x) = a x \;M_{\frac{1-c}{2}, \frac{3}{2}} \left(\frac{x^2}{2}\right)
+ b x\; U_{\frac{1-c}{2}, \frac{3}{2}}\left( \frac{x^2}{2}\right)$$
where $M$ and $U$ are the Kummer M and U functions.
