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Differential equations by series and also by an elementary method

  1. Feb 18, 2013 #1
    1. The problem statement, all variables and given/known data

    Solve the following differential equations by series and also by an elementary method and verify that your solutions agree.

    [tex](x^2+2x)y''-2(x+1)y'+2y=0[/tex]

    2. Relevant equations

    [tex]y=\sum_{n=0}^{\infty} a_nx^n[/tex]
    [tex]y'=\sum_{n=1}^{\infty} na_nx^{n-1}[/tex]
    [tex]y''=\sum_{n=1}^{\infty} n(n+1)a_{n+1}x^{n-1}[/tex]

    3. The attempt at a solution

    I have got [tex]a_1=a_0,\ a_3=0,\ a_4=-\frac{1}{8}a_3=0,\ a_5=-\frac{1}{5}a_4=0[/tex]. Then, how do we find the recursive relation to find the solution?
     
    Last edited: Feb 18, 2013
  2. jcsd
  3. Feb 18, 2013 #2

    vela

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    Isn't x=0 a singular point? You need to use the Frobenius method to get the series solution.

    For the recursion relation, set the coefficient of xn to 0.
     
    Last edited: Feb 18, 2013
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