Chapter 1.2, Problem 51P

(a) Derive the polynomial of degree five that satisfies both the Legendre equation

$\left(1-x^2\right)y^{″}-2x{y}^{\prime }+30y=0$

and the normalization condition $y\left(1\right)=1$.

(b) Sketch your solution from (a) and determine approximations to all zeros and local maxima and local minima on the interval $\left(-1,1\right)$.