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Solve Diff. Eq. using power series

  1. Mar 19, 2017 #1
    1. The problem statement, all variables and given/known data
    \begin{equation}
    (1-x)y^{"}+y = 0
    \end{equation}

    I am here but do not understand how to combine the two summations:
    Mod note: Fixed LaTeX in following equation.
    $$(1-x)\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n+\sum_{n=0}^{\infty}a_nx^n = 0$$
     
    Last edited by a moderator: Apr 1, 2017
  2. jcsd
  3. Mar 19, 2017 #2

    Ray Vickson

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    Here is your equation using PF-compatible TeX:
    $$
    (1-x)\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2} x^n+\sum_{n=0}^{\infty}a_nx^n = 0
    $$
    Just replace the "\ begin {equation} ... \ end {equation} " by "$ $ ... $ $ " (no spaces between the initial and final $ signs). Also: write "\infty", not "\infinity".

    As for your question: write out the first 3 or 4 terms, to see what you get. That will give you insight into what you should do next.
     
    Last edited by a moderator: Apr 1, 2017
  4. Mar 19, 2017 #3
    Thanks for the response and Latex help. Writing out the first few terms of each sum:
    $$
    (1-x)[2a_2+6a_3x+12a_4x^2+...]+[a_0+a_1x+a_2x^2+...]
    $$
    I am not sure what to do with the (1-x) term outside the first sum...
     
    Last edited: Mar 19, 2017
  5. Mar 19, 2017 #4

    Ray Vickson

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    What is preventing you from "distributing out" the product? That is, ##(1-x) P(x) = P(x) - x P(x).##
     
    Last edited: Mar 20, 2017
  6. Mar 31, 2017 #5
    Been a while since I did DE, but doesn't the OP have to be aware of the singular point in this problem? Hence he has to use the method of Frobenius?
     
  7. Apr 1, 2017 #6

    LCKurtz

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    But the singular point isn't at ##x=0##.
     
  8. Apr 2, 2017 #7
    Thanks for the correction. It's been a while. I remembered that there is no singular point if we take the Taylor expansion about x=0? Correct?
     
  9. Apr 2, 2017 #8

    epenguin

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    Practically after your first equation, or at any later stage, just multiply it out. You have shown that you know how to write a sum of different powers of x in terms of xn by changing the subscript appropriately.
     
  10. Apr 2, 2017 #9

    vela

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    That's backwards. If you expand about a regular point, then the solution can be written as a Taylor series. If there's a singular point, then you can use the method of Frobenius and end up with a Laurent series.
     
  11. Apr 2, 2017 #10
    Thanks Vela!

    It has been a while. I may pop open a differentials to bring the memory back.
     
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