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ODE: Finite Zeroes if k < 1/4

  1. Dec 10, 2013 #1
    Consider [itex]\frac{d^{2}y}{dx^{2}}+\frac{k}{x^{2}}y = 0[/itex]. Show that every nontrivial solution has an infinite number of positive zeroes if k > 1/4 and a finite number if k ≤ 1/4.

    Solving gives:
    [itex]y = Asin(\sqrt{k}ln(x)) + Bcos(\sqrt{k}ln(x))[/itex]​

    And setting y = 0 gives:
    [itex]tan(\sqrt{k}ln(x)) = -\frac{A}{B} = c[/itex]
    [itex]ln(x) = \frac{2n+1}{\sqrt{k}}arctan(c)[/itex]​

    So I've mainly just rearranged everything a few times. When k ≤ 1/4, the last equation should hold for only a finite number of ns. Why, though? How would one go about proving this? The main thing I would think to look for would be a square root turning negative, but that doesn't seem to happen anywhere. Any ideas?
     
  2. jcsd
  3. Dec 10, 2013 #2

    Mark44

    Staff: Mentor

    Show what you did to get this. I believe you might be tacitly making assumptions about k to get to this point.
     
  4. Dec 11, 2013 #3

    vela

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    Your y(x) doesn't satisfy the differential equation.
     
  5. Dec 11, 2013 #4
    Sorry, I did a sloppy substitution. x = e^z gives y'' - y' + ky = 0, which has the needed 1 - 4k discriminate. Problem solved. Thanks.
     
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