The Legendre Equation. 22 through 29 deal with the Legendre8 equation(1 x2)y 2xy+ ( +

Chapter 5, Problem 23

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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 if is zero or a positive even integer 2n, the series solution y1 reduces to apolynomial of degree 2n containing only even powers of x. Find the polynomials correspondingto = 0, 2, and 4. Show that if is a positive odd integer 2n + 1, the seriessolution y2 reduces to a polynomial of degree 2n + 1 containing only odd powers of x.Find the polynomials corresponding to = 1, 3, and 5.