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Math Help - Power Series Solutions of ODE's

  1. #1
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    Power Series Solutions of ODE's

    I have the following question that i am really stuck on please help.

    consider the differential equation (1+x^2)y''-6y=0.
    show that x=0 is an ordinary point of this equation - which i have done.

    determine the interval in x for which the equation has a convergent solution of the form y=sigma(akx^k).

    Show that the recurrence relation is
    ak+2=-[(k-3)/(k+1)]*ak

    and hence, determine the first six non-zero terms of the series solution.

    please help.
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by nahal View Post
    I have the following question that i am really stuck on please help.

    consider the differential equation (1+x^2)y''-6y=0.
    show that x=0 is an ordinary point of this equation - which i have done.

    determine the interval in x for which the equation has a convergent solution of the form y=sigma(akx^k).

    Show that the recurrence relation is
    ak+2=-[(k-3)/(k+1)]*ak

    and hence, determine the first six non-zero terms of the series solution.

    please help.
    these problems are a pain to type out with LaTex, so i will give you some references. tell me if they help

    see here

    and here

    and also here


    just to start you off. we are expanding around x_0 = 0, so we assume a solution of y = \sum_{k = 0}^{\infty}a_kx^k

    so, y' = \sum_{k = 1}^{\infty}ka_kx^{k - 1}

    and, y'' = \sum_{k = 2}^{\infty}k(k - 1)a_k x^{k - 2}


    now your problem is: (1 + x^2)y'' - 6y = 0

    \Rightarrow y'' + x^2y'' - 6y = 0

    now plug in the series forms for y and y'' and continue as you see in the links i gave you
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  3. #3
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    Thank you for all your help, it was well appreciated.

    There werew some very useful links that were provided.
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by nahal View Post
    Thank you for all your help, it was well appreciated.

    There werew some very useful links that were provided.
    you're welcome.

    i suppose you solved the question ok then?
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