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Power series differential equations question

  1. Aug 25, 2011 #1
    1. The problem statement, all variables and given/known data

    Using power series, find solutions to the following DE

    y' + y= x^2, y(1)= 2 and xo=1



    2. Relevant equations

    y(x)=an[itex]\sum[/itex](x-xo)^n for n=1 to infinity

    3. The attempt at a solution

    See the attachment

    NOTE: I only want to find a way to collect all the x terms in the series as one like group. Similary, I want to do the same with the coefficeints

    I will try to solve the rest myself
     

    Attached Files:

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  2. jcsd
  3. Aug 25, 2011 #2

    CompuChip

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    I think the usual way to do this is:
    [tex]y'(x) + y(x) = \sum_{n = 0}^\infty a_n n (x - x_0)^{n - 1} + \sum_{n = 0}^\infty a_n (x - x_0)^{n}[/tex]
    as you write. Now note that for n = 0, the first term of the first sum vanishes, so you can rewrite this to
    [tex]\sum_{n = 0}^\infty a_n n (x - x_0)^{n - 1} = \sum_{n = 1}^\infty a_n n (x - x_0)^{n - 1} = \sum_{n = 0}^\infty a_{n + 1} (n + 1) (x - x_0)^{n}.[/tex]
    Now you can merge the sums again:
    [tex]y'(x) + y(x) = \sum_{n = 0}^\infty b_n (x - x_0)^n[/tex]
    and derive a recursive equation for the [itex]a_n[/itex].
     
  4. Aug 25, 2011 #3
    thanks
     
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