Reduce uxx+2xuxy+(1-y^2)uyy=0 to canonical form.

Question

It's so complicated.

Comments

Plz, help me to answer that kind of question,thank you!!

Answer 1

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.