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

Answer:

Degree $k=3$

Step-by-step 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:

$x\frac{\partial s}{\partial x}+y\frac{\partial s}{\partial y}=3 s(x,y)$.

To do that let's compute the partial derivatives:

$\frac{\partial s}{\partial x}=2xy+y^2$

$\frac{\partial s}{\partial y}=x^2+2xy$

Then,

$x\frac{\partial s}{\partial x}+y\frac{\partial s}{\partial y}=2x^{2}y+xy^{2}+yx^{2}+2xy^{2}=3xy^{2}+3x^{2}y=3(x^{2}y+xy^{2})=3s(x,y).$