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

2 Answers
Nov 8, 2016

# 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

Nov 8, 2016

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#