Question

1. Determine the degree of homogeneity of the function S(x,y) = x^2 y + xy^2, and show that Euler's Theorem holds. 24 zvy

Answer 1

Degree
`k=3`

Explanation:

The function that you have is a homogeneous function of degree
`k=3`. In fact, if
`\lambda \in \mathbb{R}` and
`(x,y) \in \mathbb{R}^(2)` we have that:

`s(\lambda x, \lambda y)=(\lambda y)(\lambda x)^(2)+(\lambda x)(\lambda y)^(2)=\lambda^(3)x^(2)y+\lambda^(3)xy^(2)=\lambda^(3)s(x,y)`

To verify that the Euler Theorem holds we need to show that:

`x(\partial s)/(\partial x)+y(\partial s)/(\partial y)=3 s(x,y)`.

To do that let's compute the partial derivatives:

`(\partial s)/(\partial x)=2xy+y^2`

`(\partial s)/(\partial y)=x^2+2xy`

Then,

`x(\partial s)/(\partial x)+y(\partial s)/(\partial y)=2x^(2)y+xy^(2)+yx^(2)+2xy^(2)=3xy^(2)+3x^(2)y=3(x^(2)y+xy^(2))=3s(x,y).`