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Series solution to Second-order ODE

  1. Mar 26, 2009 #1
    1. The problem statement, all variables and given/known data

    Find the first four non-vanishing terms in a series solution of the form [tex]\sum[/tex] from 0 to infinity of akxk for the initial value problem,

    4xy''(x) + 6y'(x) + y(x) = 0, y(0) = 1 and y'(0) = -1/6

    2. Relevant equations

    3. The attempt at a solution

    Taking the second derivative of the series solution form I obtained,

    y = [tex]\sum[/tex] from 0 to infinity of akxk
    y' = [tex]\sum[/tex] from 1 to infinity of kakxk-1
    y'' = [tex]\sum[/tex] from 0 to infinity of (k+2)(k+1)ak+2xk

    Substituting into the ODE I obtained,

    4[tex]\sum[/tex] from 0 to infinity of (k+2)(k+1)ak+2xk+1 + 6[tex]\sum[/tex] from 1 to infinity of kakxk-1 + [tex]\sum[/tex] from 0 to infinity of akxk

    Now, I am unsure of where to go from here. Does this become two separate series for [tex]\sum[/tex] from 0 to infinity and [tex]\sum[/tex] from 1 to infinity?
  2. jcsd
  3. Mar 27, 2009 #2


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    Science Advisor

    No, it should be written as one series. Note that your series from 1 to infinity can just as well be written as from 0 to infinity, because kak is zero when k=0.
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