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Cauchy's homogeneous diff eqn

  1. Oct 24, 2012 #1
    The Cauchy homogeneous linear differential equation is given by

    [itex]x^{n}\frac{d^{n}y}{dx^{n}} +k_{1} x^{n-1}\frac{d^{n-1}y}{dx^{n-1}} +...+k_{n}y=X [/itex]


    where X is a function of x and [itex] k_{1},k_{2}...,k_{n}[/itex] are constants.

    I thought for this equation to be homogeneous the right side should be 0. (i.e.) X=0.
    But if X is a function of x then how can this be homogeneous?

    Thanks a lot :)
     
  2. jcsd
  3. Oct 24, 2012 #2

    HallsofIvy

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    It isn't. Why do you call it "homogeneous"?

    (Googling "Cauchy's homogeneous equation", I found a "youtube" tape calling this equation "homogeneous"- its just wrong! I suspect they started talking about a homogeneous equation and did not change the title when they generalized).
     
    Last edited: Oct 24, 2012
  4. Oct 24, 2012 #3
    Let y(x) = Y(x)+(X/kn) and the rigth side will be 0.
     
  5. Oct 24, 2012 #4
    If I do things like that, I can make any equation homogeneous.

    Don't forget that X is a function of 'x'.
     
  6. Oct 24, 2012 #5

    HallsofIvy

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    You are assuming that X is a constant, aren't you?
     
  7. Oct 24, 2012 #6
    Sorry, I was assuming that X was constant.
    So, my answer is out of subject.
     
  8. Oct 24, 2012 #7
    I have chosen that X to be a function of 'x' and it is not a constant

    So I think it is not homogeneous.

    For the record even if X is a constant it is still not homogeneous, isn't it?

    thanks a lot :)
     
  9. Oct 26, 2012 #8

    HallsofIvy

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    Yes, that is correct.
     
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