Solving a differential equation using Bessel functions

**garunas** · May 18th 2011, 02:26 PM

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?

**mr fantastic** · May 18th 2011, 07:05 PM

Originally Posted by garunas

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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