I have to solve the PDE: $$yu_{xx} + (x + y)u_{xy} + xu_{yy} = 0$$ I've found it's hyperbolic whenever $y\not=x$ and its canonical form is $u_{\phi\psi}=-\frac{1}{\psi}u_\phi$
I'm at a bit of a loss as to what to do now.. Can I set $z=u_\phi$ and then treat it as an ODE?
Yes that's what you have to do: $$u_{\phi\psi}=-\frac{1}{\psi}u_\phi$$ $$z_{\psi}=-\frac{1}{\psi}z$$ $$\psi z_{\psi}+z=0$$ $$(\psi z)'=0$$ $$\psi z=C_1(\phi)$$ Where $z=u_{\phi}$. $$\psi u_{\phi}=C_1(\phi)$$ $$u({\psi},\phi)=\dfrac {g(\phi)}{\psi}+f(\psi)$$