Chapter 17.4, Problem 12E

The solution of the initial-value problem

$x^2{y}^{″}+x{y}^{\prime }+x^{2}y=0y\left(0\right)=1y^{\prime }\left(0\right)=0$

is called a Bessel function of order 0.

(a) Solve the initial-value problem to find a power series expansion for the Bessel function.

(b) Graph several Taylor polynomials until you reach one that looks like a good approximation to the Bessel function on the interval $\left[-5,5\right]$.