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ODE - Power Series Convergence

  1. Mar 7, 2009 #1
    1. The problem statement, all variables and given/known data

    Solve
    [tex](1-4x^2)y''+34x\cdot y'-70y=0[/tex]

    2. Relevant equations

    Basically, I found the recurrence relationship to be:
    [tex] a_{n+2}=\frac{2 (-7 + n) (-5 + 2 n)}{(n+1)(n+2)}a_n}[/tex]



    Now, I solve for y1 where y1 had a_0=0 and a_1 = 1. It is a simple polynomial of degree 7.
    That was the first part.


    The second part wants me to find the radius of convergence of y2, where y2 has a_0=1 and a_1 = 0.
    In this case, all the odd coefficients are equal to 0. But I am having a hard time trying to find the radius of convergence. I actually don't have a clue here.
    I tried doing the ratio test directly on the recurrence term, but led me to 1/x^2, but that didn't work, and I think I did wrong to.

    I tried finding the pattern and trying to find what the series actually was, but couldn't. :S
    So, now I'm stuck.

    (P.S. For the patternn, I got a_n = [tex] \frac{2*(-7)(-5)(..)(2n-9)(-5)(-1)(3)(7)(4n-9)}{(2n)!}[/tex]
    But it appears to be invalid when I do the ratio test on it according to webworks.
     
  2. jcsd
  3. Mar 8, 2009 #2

    HallsofIvy

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    Staff Emeritus
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    No, the ratio test will work fine. If it led you to 1/x^2, then you have it upside down! Look at [tex]\frac{a_{2(n+1)}x^{2(n+1)}}{a_{2n}x^{2n}}= \frac{a_{2n+2}}{a_{2n}}x^2[/itex]. For what values of x is that less than 1?

     
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