In forming PDEs from a solution we proceed thus:
If we want to find a PDE which represents the family of spheres whose centres are at the $x$-axis, we do this by differentiating w.r.t. $x$ partially first, then by $y$, and eliminate the arbitrary constants. But why we are treating $z$ as a function of $x$ and $y$? And what happens if we have more arbitrary constants than variables (may be independent variables)?
Another Question:
And when we will have $\frac{\partial^2 z}{\partial x \partial y}$ term in our PDE? Of course what happens for the other way round, that if you have $\frac{\partial^2 z}{\partial x \partial y}$ in the PDE, what can you say about the solution?