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Basis question

  1. Sep 5, 2005 #1
    Can someone give suggestions for this question?

    Find a basis of solutions for the following second-order homogeneous linear equation for positive x:
    x^2y``-xy`+y=0
     
  2. jcsd
  3. Sep 5, 2005 #2

    LeonhardEuler

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    Gold Member

    Do you know what a Cauchy-Euler equation is?
     
  4. Sep 5, 2005 #3

    HallsofIvy

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    Staff Emeritus
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    Also called an "equi-potential" equation since the "power" of x is always "equal" to the order of the derivative.

    What happens if you try a solution of the form y= xr for some real number r?
     
  5. Sep 5, 2005 #4
    i don't know what a Cauchy-Euler equation is, but how'd you think of
     
  6. Sep 6, 2005 #5
    ok, i've taken a look at the Cauchy-Euler equation in the AEM textbook, but there's two things that i think are strange:

    1) for the case of double roots, the proof to the general solution only considers
    x>O... why?
    2) for the case of complex conjugate roots, there's only a general solution for all
    positive x... why?

    btw, i've been thinking all day of y=x^r... i'm still stumped over how'd you think of that?
     
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