$p$ is $dz/dx$, $q$ is $dz/dy$
I've written down the Charpit's equations:
$\frac{dp}{a}=\frac{dq}{b}=\frac{dz}{3q^2p}=\frac{dx}{q^2}=\frac{dy}{2qp}$
But i'm clueless as to what to do next. I don't know why would anyone find these enjoyable and i'm totally clueless about what these equations would mean in the real world, but anyway, i would appreciate if someone helped me out with this.