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Math Help - Power Series Problem

  1. #1
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    Power Series Problem

    Find the recurrence relation for the coefficients of a power series of y about the given point Xo given the equation.

     (4+x^2)y^{ii} -xy^i+8y=0,  Xo=0

    I am not sure what to do with the  (4+x^2)

    Thanks for any help.
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  2. #2
    Moo
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    Hello,

    it's y'', not y^{ii}


    Let y=\sum_{n=0}^\infty a_n x^n

    Hence y'=\sum_{n=1}^\infty n a_n x^{n-1}

    and y''=\sum_{n=2}^\infty n(n-1)a_n x^{n-2}


    The equation is now :
    (4+x^2) \sum_{n=2}^\infty n(n-1)a_n x^{n-2}-x \sum_{n=1}^\infty n a_n x^{n-1}+8 \sum_{n=0}^\infty a_n x^n=0

    Distribute the factors :

    \sum_{n=2}^\infty 4n(n-1)a_n x^{n-2}+\sum_{n=2}^\infty n(n-1)a_n x^n-\sum_{n=1}^\infty na_n x^n+\sum_{n=0}^\infty 8a_n x^n=0

    Change the indices so that you have x^n in all sums :

    {\color{red}\sum_{n=0}^\infty 4(n+2)(n+1) a_{n+2} x^n}+\sum_{n=2}^\infty n(n-1)a_n x^n-\sum_{n=1}^\infty na_n x^n+\sum_{n=0}^\infty 8a_n x^n=0

    -----------------------------------------------------------
    Now, change the starting n's in the sums, so that they start at the same point (namely n=2)

    \sum_{n=0}^\infty 4(n+2)(n+1) a_{n+2} x^n =4(0+2)(0+1)a_{0+2}+4(1+2)(1+1)a_{1+2} x+\sum_{{\color{red}n=2}}^\infty 4(n+2)(n+1) a_{n+2} x^n =8a_2+24a_3 x+\sum_{n=2}^\infty 4(n+2)(n+1)a_{n+2} x^n


    Similarly, we have :
    \sum_{n=1}^\infty na_n x^n=a_1 x+\sum_{{\color{red}n=2}}^\infty na_n x^n

    \sum_{n=0}^\infty 8a_n x^n=8a_0+8a_1x+\sum_{{\color{red}n=2}}^\infty 8a_n x^n

    -----------------------------------------------------------
    So we have the new equation :

    \left(8a_2+24a_3 x+\sum_{n=2}^\infty 4(n+2)(n+1)a_{n+2} x^n\right)+\left(\sum_{n=2}^\infty n(n-1)a_n x^n\right) -\left(a_1 x+\sum_{n=2}^\infty na_n x^n\right)+\left(8a_0+8a_1x+\sum_{n=2}^\infty 8a_n x^n\right)=0

    Group the terms :

    \bigg(8a_2+8a_0+x(8a_1+24a_3-a_1)\bigg) +\sum_{n=2}^\infty x^n \big(4(n+2)(n+1)a_{n+2}+n(n-1)a_n-n a_n+8a_n\big)=0

    \bigg(8a_2+8a_0+x(8a_1+24a_3-a_1)\bigg)+\sum_{n=2}^\infty x^n \big(4(n+2)(n+1)a_{n+2}+a_n [n^2-2n+8]\big)=0


    Now you'll have to use the initial conditions to find a recursive relation for a_n


    Hope that helps
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  3. #3
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    Sorry I didn't know how to make the correct "prime" sign. But thanks for the incredible help
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