I created the following example, but am unsure if my final answer is correct or in the nicest possible form. (My study materials' examples don't involve the mixed partial, and none have the integral sign in their final answers as I do below.) $\phi_1$ and $\phi_2$ are arbitrary functions.
Differentiating the final result: \begin{align}\frac{\partial u}{\partial x}&=-4x+\sqrt{x}\phi_1(4x+y^2)\\ \frac{\partial^2 u}{\partial y\partial x}&=2y\sqrt{x}\left[\phi_1'(z)\right]_{z=4x+y^2}\\ \frac{\partial^2 u}{\partial x^2}&=-4+\frac1{2\sqrt x}\phi_1(4x+y^2)+4\sqrt{x}\left[\phi_1'(z)\right]_{z=4x+y^2} \end{align} Plugging the above into the given PDE gives $-2=-2;$ therefore, that final result is indeed a solution to the PDE.
According to jjacquelin (in a comment above), this solution has no simpler form unless some boundary condition is specified.