For the equation:

$\displaystyle 2x^2u_{xx} + 5xyu_{xy} + 2y^2u_{yy} + 8xu_{x} + 5yu_y = 0$

I'm just trying to understand how to put it in canonical form.

I've understood till the point where the characteristics are:

$\displaystyle \xi = \frac{y}{x^2} $

AND

$\displaystyle \eta = \frac{y^2}{x}$

I'm just not sure how that leads to the canonical form:

$\displaystyle \eta u_{\xi\eta} + u_{\xi} = 0$