Chapter 11.7, Problem 16AP

To determine

(a)

To show:

That if $a=N$, then the given equation has a polynomial solution of degree $N$.

To determine

(b)

To show:

That the solution obtained in part (a) satisfies the expression,

${\left[\sqrt{1-x^2}{T}_N{}^{\prime }\right]}^{\prime }+\frac{N}{\sqrt{1-x^2}}{T}_N=0$

To determine

(c)

To prove:

That the integral ${\displaystyle {\int }_{-1}^1\frac{T_N\left(x\right)T_M\left(x\right)}{\sqrt{1-x^2}}dx}=0$.