ux2+2ux2y+(1-y2)uy2=0,
ux2+2ux2y+uy2-uy4=0,
x2+2x2y+y2-y4=0 (the u's cancel out by dividing each term by u).
This is one canonical form where x precedes y in the equation.
However, y has the highest degree so another form could be:
-y4+x2+2x2y+y2=0, which places the highest order term first.
Canonical or standard form always combines products of the same variable with itself, so xx becomes x2 and this applies even if the variables are separated in a product, for example, xyx is written x2y.