Chapter 12.10, Problem 1P

Verify equations (10.3) and (10.4).

(10.4) $\left(1-x^2\right)u^{\prime \prime }-2(m+1)x{u}^{\prime }+[l(l+1)-m(m+1)]u=0$.

For $m=0,$ this is Legendre’s equation with solutions $P_l(x).$ Differentiate (10.3) ,obtaining (Problem 1)

(10.4) $\left(1-x^2\right){\left(u^{\prime }\right)}^{\prime \prime }-2[(m+1)+1]x{\left(u^{\prime }\right)}^{\prime }+[l(l+1)-(m+1)(m+2)]u^{\prime }=0$.