# Solve $pq^2=ax+by$ - a nonlinear first-order pde

$$p$$ 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.

• 'I'm clueless as to what to do next'... Solve them \begin{align} dp/a &= dq/b \\ \implies bp &= aq + c_{1} \\ d(p + q)/(a + b) &= dy/2qp \\ \implies \dots \end{align} – mattos Dec 27 '19 at 0:42