Chapter 9.8, Problem 65E

Bessel Function The Bessel function of order 0 is

$J_0(x)={\displaystyle \underset{k=0}{\overset{\infty }{\sum }}\frac{{(-1)}^k{x}^{2k}}{2^{2k}{(k!)}^2}.}$

(a) Show that the series converges for all x

(b) Show that die series is a solution of the differential equation $x^2{J}_0{{}^{\prime }}^{\prime }+x{J}_0{}^{\prime }+x^2{J}_0=0$.

(c) Use a graphing utility to graph die polynomial composed of the first four terms of $J_0$

(d) Approximate ${\int }_0^1{J}_{0}dx$, accurate to two decimal places.