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

Chapter 5, Problem 26

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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.The Legendre polynomials play an important role in mathematical physics. For example,insolving Laplaces equation (the potential equation) in spherical coordinates, we encounterthe equationd2F()d2 + cot dF()d+ n(n + 1)F() = 0, 0 <