Chapter 11.9, Problem 36E

a)

To determine

To show: The function $J_1\left(x\right)={\displaystyle \sum _{n=0}^{\infty }\frac{{\left(-1\right)}^n{x}^{2n+1}}{n!\left(n+1\right)!2^{2n+1}}}$ which 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)

To determine

To show:That the ${J^{\prime }}_0\left(x\right)=-J_1\left(x\right)$.