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Homework Help: Series solution to DE about ordinary point

  1. Mar 12, 2013 #1
    1. The problem statement, all variables and given/known data

    Find two power series solutions of the DE

    (x+2)y'' + xy' - y = 0

    about the ordinary point x = 0 . Include at least first four nonzero terms for each of the solutions.

    2. The attempt at a solution

    I distributed the y'' term and substituted
    y = Ʃ0inf cnxn
    and its derivatives into the DE. I equated it to 0 and got two equations:

    2c2 - c0 = 0

    xn(cn+1*n(n+1) + cn+2*(n+1)(n+2)+ cn*(n-1)) = 0

    The weird thing is that the second equation (the recurrence relationship) has three c terms in it, although the examples shown have two. How do I get c2 and c0? After that happens should I simply solve for c1 using the recurrence relationship with n = 0?
  2. jcsd
  3. Mar 12, 2013 #2


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    Staff Emeritus
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    You seem to have dropped the factor of 2 from the 2y'' term, so your two equations are slightly wrong.

    Once ##c_0## is set, you can determine ##c_n## for ##n \ge 2## through the recurrence relation. That's your first solution.

    The coefficient ##c_1## is still arbitrary. That corresponds to your second solution.
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