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Unformatted text preview: POWER SERIES SOLUTIONS OF SECOND-ORDER LINEAR EQUATIONS KIAM HEONG KWA 1. Method of Undetermined Coefficients Consider a general second-order linear homogeneous equation (1.1) y 00 + p ( x ) y + q ( x ) y = 0 , where the coefficients are real analytic in the sense that they admit convergent power series representations (1.2) p ( x ) = X k =0 p k ( x- x ) k and q ( x ) = X k =0 q k ( x- x ) k about a point x , called the center of expansion. By a translation of coordinates, one reduces the consideration of such an equation having the origin x = 0 as the center of expansion, so that (1.3) p ( x ) = X k =0 p k x k and q ( x ) = X k =0 q k x k . Without worrying about the convergence for the moment, it will be shown by construction that there is a unique power series (1.4) y ( x ) = X k =0 a k x k that formally satisfies (1.1) for each choice of a = y (0) and a 1 = y (0). Date : February 5, 2011. 1 2 KIAM HEONG KWA To begin with the construction, one differentiates (1.4) term by term to obtain q ( x ) y ( x ) = X k =0 k X j =0 q j a k- j x k , (1.5) p ( x ) y ( x ) = X k =0 k X j =0 ( k + 1- j ) p j a k +1- j x k , y 00 ( x ) = X k =0 ( k + 1)( k + 2) a k +2 x k . Then substituting into (1.1) and equating the coefficients of the like powers of x to zero yields (1.6) a k +2 =- k j =0 [( k + 1- j ) p j a k +1- j + q j a k- j ] ( k + 1)( k + 2) , k ....
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