I have the following PDE for a function $u(t,x,y)$ defined for $t,x,y>0$: $$ 2\partial_tu=x\partial_{xx}u+\sqrt{xy}\partial_{xy}u+\frac y2\partial_{yy}u+2\rho\partial_xu+\rho\partial_yu $$ where $\rho>1$ is a constant.
Is it possible to solve this equation explicitly?