# Verifying the integral form of the Bessel equation by substitution

1. Feb 7, 2012

### tjackson3

1. The problem statement, all variables and given/known data
The following is an integral form of the Bessel equation of order n:

$$J_n(x) = \frac{1}{\pi}\int_0^{\pi}\ \cos(x\sin(t)-nt)\ dt$$

Show by substitution that this satisfies the Bessel equation of order n.

2. Relevant equations

Bessel equation of order n: $x^2y'' + xy' + (x^2-n^2)y = 0$

3. The attempt at a solution

I tried simply plugging this into the Bessel equation, but it didn't really help. You end up with:

$$\int_0^{\pi} \left[-x^2\sin^2 t\cos(x\sin t - nt) - x\sin t\sin(x\sin t - nt) + (x^2-n^2)\cos(x\sin t - nt)\right]\ dt = 0$$

This is all assuming you can interchange the order of integration and differentiation, but it seems like you should be able to, since the integration is wrt t, while the differentiation is wrt x.

Thanks!