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Math Help - Homogeneity and Proving a Function's Homogeneity

  1. #1
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    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:

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

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

    f(x,y) = 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!
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Homogeneity and Proving a Function's Homogeneity

    Quote Originally Posted by Number42 View Post
    The function is: f(x,y) = x + ye^y^/^x .
    f(tx,ty)=tx+tye^{ty/tx}=t^1f(x,y)


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

    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
    Last edited by FernandoRevilla; November 7th 2011 at 02:19 AM.
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