Use characteristics to obtain the canonical form of an equation

I was asked to consider this parabolic equation.

$ 3 frac {∂ ^ 2u} {∂x ^ 2} + 6 frac {∂ ^ 2u} {∂x∂y} +3 frac {∂ ^ 2u} {∂y ^ 2} – frac {∂ u} {∂x} – 4 frac {∂u} {∂y} + u = 0 $

I've calculated the characteristic coordinates to be $ ξ = y – x, η = x $. The question then asks to transform the equation into a canonical form. I have the method in other questions, but I do not understand how to transfer the method of these examples to this one.