Follow the steps outlined in this problem to establish Roudrigues’s formulafor the nth-deg...

Follow the steps outlined in this problem to establish Roudrigues’s formula

for the nth-degree polynomial. (a) show that υ = (x2 − 1)n satisfies the differential equation.

Differentiate each side of this equation to obtain

Differentiate each side of last equation n times in succession to obtain

Thus u = υ(n) = Dn(x2 − 1)n satisfies Legendre’s equation of order n. (c) Show that the coefficient of xn in u is (2n)!/n!; then state why this proves Rodrigues′ formula. (Note that the coefficient of xn in Pn(x) is (2n)!/ [2n(n!)2].)