I am looking for examples of exact known simple solutions to the second order elliptic PDE $$a_{11}(x)\partial_x^2u+a_{22}(x)\partial_y^2u+2a_{12}(x)\partial_{xy}^2u+ b_1(x)\partial_1u+b_2(x)\partial_2u=0\quad x\in D$$ where $D\subseteq\mathbb{R}^2$ is an open set. Are tons of examples for $\Delta u=0$, but I couldn't find examples with nonconstant $a(x)$ and nonzero $b(x)$. I need at least 3 examples.
EDIT: I found one: $u(x,y)=e^{xy}$ satisfies $$\left(\partial_1^2+\partial_2^2-2\partial_{12}+(x-y)(\partial_1-\partial_2)\right)u=0 $$
Just take any smooth $u(x,y)$, any $a_{ij}$ that satisfy the ellipticity conditions, and choose $b_1, b_2$ so that the equation holds. It helps to make sure at least one of $\partial_x u, \partial_y u$ is nowhere vanishing. Here is a random example, with no attempt to make things look nice:
$$u(x,y) = \sin x + e^y$$ $$a_{11} = 4+x^2, \quad a_{12} = e^{-x^2-y^2},\quad a_{22} = 1+y^2, \quad b_1 = \cos(xy)$$ and now you can solve for $b_2$ from $$ (4+x^2)(-\sin x) + (1+y^2) e^{y} + \cos(xy) \cos x + b_2 e^y = 0 $$ obtaining $$b_2 = -e^{-y}\left[(4+x^2)(-\sin x) + (1+y^2) e^{y} + \cos(xy) \cos x\right]$$