Chapter 9.8, Problem 65E

Bessel FunctionThe 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 tire series is a solution of the differential equation $x^2{J}_0^n+x{J}_0^{\text{'}}+x^2{J}_0=0$.

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

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