Chapter 1.2, Problem 26P

When $N$ is a positive integer, the Legendre equation

$\left(1-x^2\right)y^{″}-2x{y}^{\prime }+N\left(N+1\right)y=0$,

With $-1<x<1$, has a solution that is a polynomial of degree $N$. Show by substitution into the differential equation that in the case $N=3$ such a solution is

$y\left(x\right)=\frac{1}{2}x\left(5x^2-3\right)$