Solution to a second order PDE

Question

Looking for a general solution to the following second-order PDE, where the unknown is a function $f(x_1, x_2)$ of two variables:

$ 0=a^2f+a^2x_1{\partial f\over \partial x_1}+b^2x_2{\partial f\over \partial x_2}+\frac{1}{2} \left(a^2{\partial^2 f\over \partial x_1^2} + 2ab{\partial^2 f\over \partial x_1\partial x_2}+b^2{\partial^2 f\over \partial x_2^2}\right) $

where $a,b$ are positive reals.

(Note that this may be seen as a 2D generalization of the Hermite differential equation, by setting $b=0$.)

Thanks!

Answer 1

If you take the Fourier transform, the equation reduces to a first order hyperbolic equation, which can then be solved by the method of characteristics.