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Math Help - Infinitely differentiable and homegeneous function

  1. #1
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    Infinitely differentiable and homegeneous function

    Suppose f is a function such that it is infinitely differentiable and homegeneous of degree n \in \mathbb{N},show that

    x^{2}\frac{\partial^{2}f}{\partial x^{2}}+2xy\frac{\partial^{2}f}{\partial x\partial y}+y^{2}\frac{\partial^{2}f}{\partial y^{2}}=n(n-1)f(x,y)

    Hi all,

    for the above question, what is the link of f' in terms of f and n?? I tried using the rate of change quotient here, but I cant get the prove done. I think that if i can find f' in terms of f and n, then f'' should be far. Thanks.
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  2. #2
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    I found out this is solved based on Euler Theorem already. (in case anyone else is interested to learn too!)
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  3. #3
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    I dont think it is necessary to use Euler's Theorem. Just differentiate the equation f(tx,ty)=(t^n)f(x,y) twice with respect to t.
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