Chapter 5.7, Problem 9P

(a)

To determine

To show: That the equation $L\left[y_2\right]\left(x\right)={\displaystyle \sum _{n=1}^{\infty }n\left(n-1\right)b_n{x}^n+{\displaystyle \sum _{n=1}^{\infty }n{b}_n{x}^n+{\displaystyle \sum _{n=1}^{\infty }{b}_n{x}^{n+2}+2}}x{J}_0^{\text{'}}\left(x\right)}$.

(b)

To determine

To show: That $b_{1}x+2^2{b}_2{x}^2+{\displaystyle \sum _{n=1}^{\infty }\left(n^2{b}_n+b_{n-2}\right)x^n=-2{\displaystyle \sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n}2n{x}^{2n}}{2^{2n}{\left(n!\right)}^2}}}$ on substituting the series representation for $J_0\left(x\right)$ in part (a) equation.

(c)

To determine

To show: That $b_1=b_3=b_5=\cdot \cdot \cdot =0,b_2=\frac{1}{2}{\left(1!\right)}^2$, ${\left(2n\right)}^2{b}_{2n}+b_{2n-2}=-2{\left(-1\right)}^n\left(2n\right)/2^{2n}\left(n\right)!$ and deduce $b_4=-\frac{1}{2^2{4}^2}\left(1+\frac{1}{2}\right)\text{ and }{b}_6=\frac{1}{2^2{4}^2{6}^2}\left(1+\frac{1}{2}+\frac{1}{3}\right)$.