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Find power series representations of the general solution

  1. Apr 22, 2015 #1
    1. The problem statement, all variables and given/known data

    (1+x2) y'' + 2xy' = 0 in powers of x

    2. Relevant equations

    [itex] y'' = \sum_{n=2}^{\infty} (n-1)na_nx^{n-2} [/itex]

    [itex] y' = \sum_{n=1}^{\infty} na_nx^{n-1} [/itex]

    3. The attempt at a solution

    (1+x2) y'' + 2xy' =

    [itex] (1+x^2) \sum_{n=2}^{\infty} (n-1)na_nx^{n-2} + 2x \sum_{n=1}^{\infty} na_nx^{n-1} = 0 [/itex]

    [itex] (1+x^2) \sum_{n=2}^{\infty} (n-1)na_nx^{n-2} + 2x \sum_{n=2}^{\infty} (n-1)na_nx^{n} + 2
    \sum_{n=1}^{\infty} na_nx^{n} = 0 [/itex]

    [itex] 2a_2 + 6a_3x + 2a_1x + \sum_{n=2}^{\infty} [(m+1)(m+2)a_mx + (m-1)ma_m + 2ma_m] x^{m} [/itex]

    [itex]a_0 = a_0 \\
    a_1 = a_1 \\
    6a_3 + 2a_1 = 0 \\
    12a_4 + 6a_2 = 0, a_4 = 0 \\
    20a_5 + 12a_3 = 0 \\
    [/itex]

    [itex] a_{2n} = 0, a_{2n+1} = (-1)^n\frac{a_1}{2n+1} [/itex]
     
  2. jcsd
  3. Apr 23, 2015 #2

    NascentOxygen

    User Avatar

    Staff: Mentor

    Hi @Shackleford . We're all waiting for the other shoe to drop ...... :wink:

    ... is there a question coming?
     
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