Chapter 8.6, Problem 36E

The Bessel function of order l is defined by

$J_1\left(x\right)={\displaystyle \underset{n=0}{\overset{\infty }{\sum }}\frac{{\left(-1\right)}^n{x}^{2n+1}}{n!\left(n+1\right)!2^{2n+1}}}$

(a) Show that J₁ satisfies the differential equation

$x^2{J^{″}}_1\left(x\right)+x{J^{\prime }}_1\left(x\right)+\left(x^2-1\right)J_1\left(x\right)=0$

(b) Show that ${J^{\prime }}_0\left(x\right)=-J_1\left(x\right)$