Canonical form 2nd order PDE

Question

I want to reduce the following equation to canonical form

$yu_{xx} + 2(x+y)u_{xy} + 4xu_{yy} = 0$ for $x > y > 0$

I chose ɛ to be $x^2 - \frac{y^2}{2}$ and η to be $2x - y$

Then I found the second derivatives:

$u_{xx} = 4x^2u_{ɛɛ} + 8xu_{ɛη} + 4u_{ηη}$

$u_{xy} = -2xyu_{ɛɛ} + u_{ɛη}(-2x - 2y) - 2u_{ηη}$

$u_{yy} = -y^2u_{ɛɛ} + 2yu_{ɛη} + u_{ηη}$

then when I subbed it into the original equation, most terms cancel out and I get $= u_{ɛη}(-4x^2 + 8xy - 4y^2)$

but then what? is that the canonical form? If it isn't, could someone point me in the right direction? Thank you :)

Answer 1

If $u_{\varepsilon \eta} \times \text{something} = 0$, then $u_{\varepsilon \eta} = 0$ or $\text{something} = 0$. If you go for the first option, you'll get the solution.

$$u_{\varepsilon \eta} = 0 \implies u(\varepsilon,\eta) = f(\varepsilon) + g(\eta),$$

for some arbitrary functions $f$ and $g$. Can you show us the intermediate steps?

Cheers!