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!



Solution to a second order PDE
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.


