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Homogeneous function

Function f (x, y) is a homogeneous expression if f (tx, ty) = (t^n) * f (x, y), for t (≠0). Prove that a homogeneous expression f (x, y) satisfies the following partial differential equation where p is a natural number .

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Replies

  • circularsquare
    circularsquare
    There is an alternative definition of homogeneous function of order n.
    I.e f(x,y) = (x^n)u(y/x) where u is a function of y/x.

    Example :- consider f(x,y) = x^2 + xy + y^2 you can write it as
    (x^2) (1 + y/x + (y/x)^2 ).

    Use this property to prove for case p = 1. Then use mathematical induction on p.

    Proving it is beyond my ability as there are many calculations. 😀 . Keep trying.
  • Aruwin
    Aruwin
    circularsquare
    There is an alternative definition of homogeneous function of order n.
    I.e f(x,y) = (x^n)u(y/x) where u is a function of y/x.

    Example :- consider f(x,y) = x^2 + xy + y^2 you can write it as
    (x^2) (1 + y/x + (y/x)^2 ).

    Use this property to prove for case p = 1. Then use mathematical induction on p.

    Proving it is beyond my ability as there are many calculations. 😀 . Keep trying.
    I hate doing proving problems but I am gonna try it anyway. Well, can you look at my other thread? I guess that's easier than this.
  • PraveenKumar Purushothaman
    PraveenKumar Purushothaman
    By any chance is this a homework?
  • Aruwin
    Aruwin
    Praveen-Kumar
    By any chance is this a homework?
    Nope,not homework. I have never posted any homework on here. If it was homework, I would leave it blank and wait until the teacher discuss it in class.
  • PraveenKumar Purushothaman
    PraveenKumar Purushothaman
    Okay... 😀
  • Aruwin
    Aruwin
    Praveen-Kumar
    Okay... 😀
    And as to why I always post questions on here is because people here explain better than the teachers at my university do.
  • PraveenKumar Purushothaman
    PraveenKumar Purushothaman
    Aruwin
    And as to why I always post questions on here is because people here explain better than the teachers at my university do.
    Ah! That's nice... Sure, we would be able to help you out... 😀

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