Consider the differential equation (x2 1)y + [1 (a + b)]x y + aby = 0, (11.7.9) where a

Chapter 11, Problem 15

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Consider the differential equation (x2 1)y + [1 (a + b)]x y + aby = 0, (11.7.9) where a and b are constants. (a) Show that the coefficients in a series solution to Equation (11.7.9) centered at x = 0 must satisfy the recurrence relation an+2 = (n a)(n b) (n + 2)(n + 1) an, n = 0, 1,..., and determine two linearly independent series solutions. (b) Show that if either a or b is a nonnegative integer, then one of the solutions obtained in (a) is a polynomial. (c) Show that if a is an odd positive integer and b is an even positive integer, then both of the solutions defined in (a) are polynomials. (d) If a = 5 and b = 4, determine two linearly independent polynomial solutions to Equation (11.7.9). Notice that in this case the radius of convergence of the solutions obtained is R = , whereas Theorem 11.2.1 only guarantees a radius of convergence R 1. 1