This is just a comment on Rafa's answer, but I don't seem to find the way to make comments. The PDE is not necessarily elliptic (take e.g. A = 1 and B = -1, resulting in the wave equation).

This is just a comment on Rafa's answer, but I don't seem to find the way to make comments. The PDE is not necessarily elliptic (take e.g. A = 1 and B = -1, resulting in the wave equation). If A and B are (identically) zero then the equation just says that the divergence of the vector field $Y(y, z)$ with entries $Y_1(y,z) = (Cy - Dz)P(y, z)$ and $Y_2(y, z) = (Dy + Cz)P(y, z)$ is zero. If the domain is simply connected, then that is equivalent to saying that there exists a scalar function (a potential) $V$ such that $Y = \nabla^{\perp} V$.