Here is the definition of the Bessel function I am starting with a definition as an integral.
$$ J_n(x) = \frac{1}{2\pi} \int_{-\pi}^\pi e^{i n t - x \sin t} \, dt $$
Essentially we have computed the $n$-th Fourier coefficient of a certain function:
$$ e^{-ix \sin t} = \sum e^{int} J_n(x) $$
How can I show the integral satisfies the Bessel differential equation? And which theorem helps to justify differentiating under the integral sign?
$$ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0 $$