# 3 4 12 1 x x4 2 4 a1 x 13 1 x x5 3 5 we

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Unformatted text preview: e ﬁrst two coeﬃcients a0 and a1 in the power series can be determined from the initial conditions, dy (0) = a1 . dx y (0) = a0 , Then the recursion formula can be used to determine the remaining coeﬃcients by the process of induction. Indeed it follows from (1.5) with n = 0 that a2 = − a0 1 = − a0 . 2·1 2 Similarly, it follows from (1.5) with n = 1 that a3 = − a1 1 = − a1 , 3·2 3! and with n = 2 that a4 = − a2 1 11 = a0 = a0 . 4·3 4·32 4! Continuing in this manner, we ﬁnd that a2n = (−1)n a0 , (2n)! a2n+1 = 8 (−1)n a1 . (2n + 1)! Substitution into (1.4) yields y = a0 + a1 x − = a0 1 − 1 1 1 a0 x2 − a1 x3 + a0 x4 + · · · 2! 3! 4! 12 1 x + x4 − · · · 2! 4! + a1 x − 13 1 x + x5 − · · · 3! 5! . We recognize that the expressions within parentheses are power series expansions of the functions sin x and cos x, and hence we obtain the familiar expression for the solution to the equation of simple harmonic motion, y = a0 cos x + a1 sin x. The method we have described—assuming a solution to the diﬀerential eq...
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## This document was uploaded on 01/12/2014.

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