Get solution: The Legendre Equation. 22 through 29 deal with the Legendre8 equation(1

Chapter 5, Problem 28

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The Legendre Equation. 22 through 29 deal with the Legendre8 equation(1 x2)y 2xy+ ( + 1)y = 0.As indicated in Example 3, the point x = 0 is an ordinary point of this equation, and the distancefrom the origin to the nearest zero of P(x) = 1 x2 is 1. Hence the radius of convergence ofseries solutions about x = 0 is at least 1. Also notice that we need to consider only > 1because if 1, then the substitution = (1 + ), where 0, leads to the Legendreequation (1 x2)y 2xy+ ( + 1)y = 0.Show that the Legendre equation can also be written as[(1 x2)y]= ( + 1)y.Then it follows that[(1 x2)Pn(x)]= n(n + 1)Pn(x) and [(1 x2)Pm(x)]= m(m + 1)Pm(x).By multiplying the first equation by Pm(x) and the second equation by Pn(x), integratingby parts, and then subtracting one equation from the other, show that11Pn(x)Pm(x) dx = 0 if n = m.This property of the Legendre polynomials is known as the orthogonality property. Ifm = n, it can be shown that the value of the preceding integral is 2/(2n + 1).