Answer: In each of 13 through 16, solve the given differential equation by a series in

Chapter 5, Problem 16

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In each of 13 through 16, solve the given differential equation by a series in powers of x and verify that a0 is arbitrary in each case. involves a nonhomogeneous differential equation to which series methods can be easily extended. Where possible, compare the series solution with the solution obtained by using the methods of Chapter 2. y_ y = x2 The Legendre Equation. 17 through 23 deal with the Legendre8 equation (1 x2) y__ 2xy_ + ( + 1) y = 0. As indicated in Example 4, the point x = 0 is an ordinary point of this equation, and the distance from the origin to the nearest zero of P( x) = 1 x2 is 1. Hence the radius of convergence of series solutions about x = 0 is at least 1. Also notice that we need to consider only > 1 because if 1, then the substitution = (1 + ), where 0, leads to the Legendre equation (1 x2) y__ 2xy_ + ( + 1) y = 0.