The function

$\displaystyle $J_{1} (x) = \sum_{n=0}^{\infty} (-1)^{n}\frac{x^{2n+1}}{2^{2n+1} (n!)(n+1)!}$$

is called a Bessel function of order 1. Show that it satisfies the differential equation

$\displaystyle x^{2} $$\displaystyle J''_{1}(x) $ $\displaystyle +xJ'_{1}(x) $ $\displaystyle + (x^{2}$ $\displaystyle - 1)J_{1}(x) = 0$

I have tried differentiating the series and plugging in but I cannot get the right answer.