# Homogeneity and Proving a Function's Homogeneity

• Nov 6th 2011, 08:22 PM
Number42
Homogeneity and Proving a Function's Homogeneity
Hi.

I am presented with a function for which I know is homogeneous of degree 1. I'm having trouble proving it, however.

The function is:

$\displaystyle f(x,y) = x + ye^y^/^x .$

I also have to prove that if this is homogeneous of degree 1, then

f(x,y) = $\displaystyle x(df/dx) + y(df/dy)$

which I know utilizes the chain rule, but how do I start it? I have no clue.
Thanks for any help!
• Nov 6th 2011, 11:21 PM
FernandoRevilla
Re: Homogeneity and Proving a Function's Homogeneity
Quote:

Originally Posted by Number42
The function is: $\displaystyle f(x,y) = x + ye^y^/^x .$

$\displaystyle f(tx,ty)=tx+tye^{ty/tx}=t^1f(x,y)$

Quote:

I also have to prove that if this is homogeneous of degree 1, then f(x,y) = $\displaystyle x(df/dx) + y(df/dy)$
One way (in this case):

$\displaystyle x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=x\left(1+ye^{y/x}\cdot \frac{-y}{x^2}\right)+\ldots$