Chapter 12.4, Problem 1P

Verify equations (4.4) and (4.5).

(4.4) $\left(x^2-1\right)\frac{d^{l+2}v}{d{x}^{l+2}}+(l+1)(2x)\frac{d^{l+1}v}{d{x}^{l+1}}+\frac{(l+1)l}{2!}\cdot 2\cdot \frac{d^{l}v}{d{x}^l}$ $=2lx\frac{d^{l+1}v}{d{x}^{l+1}}+2l(l+1)\frac{d^{l}v}{d{x}^l}$.

Simplifying (4.4), we get (Problem 1)

(4.5) $\left(1-x^2\right){\left(\frac{d^{l}v}{d{x}^l}\right)}^{\prime \prime }-2x{\left(\frac{d^{l}v}{d{x}^l}\right)}^{\prime }+l(l+1)\frac{d^{l}v}{d{x}^l}=0$.