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Math Help - Solving a differential equation using Bessel functions

  1. #1
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    Solving a differential equation using Bessel functions

    Let n be a positive integer. The equation:

    y'' + \frac{1}{x} y' + \left(1 - \frac{n^2}{x^2}\right)y = 0 \hspace{5mm}(1)

    has a solution which is the Bessel function of order n:

    J_{n}(x) = x^n\sum_{k=0}^\infty a_{k}x^{k} \hspace{5mm} (2)

    By substituting (2) into (1) determine a_{1}

    I've done the substitution and dont seem to be getting anywhere. Any help anyone?
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  2. #2
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    Quote Originally Posted by garunas View Post
    Let n be a positive integer. The equation:

    y'' + \frac{1}{x} y' + \left(1 - \frac{n^2}{x^2}\right)y = 0 \hspace{5mm}(1)

    has a solution which is the Bessel function of order n:

    J_{n}(x) = x^n\sum_{k=0}^\infty a_{k}x^{k} \hspace{5mm} (2)

    By substituting (2) into (1) determine a_{1}

    I've done the substitution and dont seem to be getting anywhere. Any help anyone?
    Please show all your work.
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