Reduce the following second order PDE to its canonical form

Question

I got this PDE: $u_{xx} + xu_{yy}=0$ on the domains $D_{-}=\{(x,y): x<0\},D_{+}=\{(x,y): x>0\}. $ I see that the PDE is hyperbolic on $D_{-}$ and ellipctic on $D_{+}.$ I'm having trouble finding its canonical form though. Any help, with a bit of detail would be greatly appreciated!

Answer 1

The characteristics are given by the equation

$$ \frac{\mathrm{d}y}{\mathrm{d}x} = \frac{-B \pm \sqrt{B^2-4AC}}{2A},$$

where $A = 1$, $B = 0$, and $C = x$. Substituting and integrating gives the two complex characteristics $\xi = \frac{2i}{3}x^{3/2}+y$, and $\eta = -\frac{2i}{3} x^{3/2}+y$. By using the chain rule, i.e., recalling that $\partial_x = \xi_x \partial_\xi + \eta_x \partial_\eta$ and $\partial_y = \xi_y \partial_\xi + \eta_y \partial_\eta$ one arrives at the canonical form

$$ 6 (\eta-\xi) u_{\xi\eta} + u_\xi - u_\eta = 0.$$

I hope you find this useful.