The Legendre Equation. 22 through 29 deal with the

Chapter 5, Problem 22

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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 two solutions of the Legendre equation for |x| < 1 arey1(x) = 1 ( + 1)2! x2 + ( 2)( + 1)( + 3)4! x4+ m=3(1)m ( 2m + 2)( + 1)( + 2m 1)(2m)! x2m,y2(x) = x ( 1)( + 2)3! x3 + ( 1)( 3)( + 2)( + 4)5! x5+ m=3(1)m ( 1)( 2m + 1)( + 2)( + 2m)(2m + 1)!

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