Question

How do you determine if `f(x,y)=-x^3-4xy^2+y^3` is homogeneous and what would it's degree be?

Answer 1

`f(x,y)` is a homogeneous function of degree 3

Explanation:

`f(x,y) = -x^3 - 4xy^2 + y^3`

Each term in `f(x,y)` is a power of `x` and `y` only, and the sum of the powers of each term is the same (in this case 3). This meets the definition of a homogeneous function.

Hence `f(x,y)` is a homogeneous function of degree 3

Answer 2

See below.

Explanation:

A Homogeneous function, is a function which obeys the relationship

`f(lambda x_1,lambda x_2,cdots,lambda x_n)=lambda^nf(x_1,x_2,cdots, x_n)`

In the present case we have

`f(lambda x, lambda y) = -(lambda x)^3-2(lambdax)(lambda y)^2+(lambday)^3=lambda^3(-x^3-xy^2+y^3)=lambda^3f(x,y)`

so `f(x,y)` is homogeneous with degree `3`