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How do you determine if #f(x,y)=-x^3+3x^2y^2-2y^2# is homogeneous and what would it's degree be?

Answer 1

You want to check that #P(lambda x, lambda y) =lambda^n P(x,y)# That can not be the case since P is the sum of three terms of unequal degree. So the answer is no.

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Answer 2

Isabella Ackerman

A function ( f(x, y) = -x^3 + 3x^2y^2 - 2y^2 ) is homogeneous if it satisfies the condition ( f(tx, ty) = t^n f(x, y) ) for all ( x, y, t ) such that ( t \neq 0 ), where ( n ) is the degree of homogeneity. To determine its degree:

Substitute ( tx ) and ( ty ) into the function.
Simplify both sides and determine if they are equal.
If they are equal, find the value of ( n ).

In this case:

[ f(tx, ty) = - (tx)^3 + 3(tx)^2(ty)^2 - 2(ty)^2 ]

Simplify and compare this to ( t^n f(x, y) ). If they are equal, then ( f(x, y) ) is homogeneous of degree ( n ).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.