Chapter 5.3, Problem 29P

To determine

To show: From the result of The Legendre equation ${\displaystyle {\int }_{-1}^1{P}_n\left(x\right)P_m\left(x\right)}dx=\frac{2}{2n+1}\text{ if }n=m$ is ${\left[\left(1-x^2{y}^{\prime }\right)\right]}^{\prime }=-\alpha \left(\alpha +1\right)y$ for the polynomial $f\left(x\right)={\displaystyle \sum _{k=0}^n{a}_n}{P}_k\left(x\right)$.