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Math Help - partial differential equation

  1. #1
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    partial differential equation

    A function f of two variables is said to be homogeneous of degree n if f(tx,ty)=t^nf(x,y) whenever t>0.

    Show that such a function f satisfies the equation x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=nf
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  2. #2
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    Quote Originally Posted by vuze88 View Post
    A function f of two variables is said to be homogeneous of degree n if f(tx,ty)=t^nf(x,y) whenever t>0.

    Show that such a function f satisfies the equation x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}=nf

    Derive both sides of f(tx,ty)=t^nf(x,y) wrt t and then evaluate the result at t=1 .

    Tonio
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  3. #3
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    Quote Originally Posted by tonio View Post
    Derive both sides of f(tx,ty)=t^nf(x,y) wrt t and then evaluate the result at t=1 .

    Tonio
    Sorry i still dont fully understand. Can you explain this in more detail. Also im not sure whether the f on the RHS means f(x,y) or f(tx,ty)
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  4. #4
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    Quote Originally Posted by vuze88 View Post
    Sorry i still dont fully understand. Can you explain this in more detail. Also im not sure whether the f on the RHS means f(x,y) or f(tx,ty)

    It is exactly as I wrote it: derivate the left side wrt t, derivate the right side wrt t, they're equal, of course, and now put t=1 .

    Tonio
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  5. #5
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    lol yeah but how to i differentiate with respect to t when the function is in terms of tx and ty
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  6. #6
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    Use the chain rule, of course!

    \frac{df(ax)}{dx}= a\frac{df}{dx}
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  7. #7
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    Euler's theorem

    Quote Originally Posted by HallsofIvy View Post
    Use the chain rule, of course!

    \frac{df(ax)}{dx}= a\frac{df}{dx}
    this is what Euler's theorem on partial derivatives says.
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