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Unformatted text preview: Homework Solutions: Problem 7 e) in 4.2 The equation is xy 00- 2 y + xy = 0. The first problem is to obtain the radius of convergence of the series solution about the point x = 1. To do this first write the equation in standard form y 00- 2 x y + y = 0 . In the notation of the text p ( x ) =- 2 /x and q ( x ) = 1. To answer the question it is necessary to express p and q in power series about x = 1 and determine their radii of convergence. q , being a constant, is trivially a power series whose radius of convergence is R q = . For p , we use the geometric series 1 / (1 + y ) = (- 1) n y y :- 2 x =- 2 1 + ( x- 1) =- X n =0 2(- 1) n ( x- 1) n = X n =0 2(- 1) n +1 ( x- 1) n . This converges if and only if | x- 1 | < 1 and hence its radius of convergence is R p = 1. The smaller of the two radii of convergence is 1; therefore the representation of a solution y ( x ) to xy 00- 2 y + xy = 0 as a power series y ( x ) = a n ( x- 1) n centered at x = 1 will have at least a radius of 1....
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- Fall '06