6 and so forth thus we nd that a6 y a0 1 2p 2 22 pp

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Unformatted text preview: ecursion formula for the coefficients of the power series: an+2 = 2n − 2p an . (n + 2)(n + 1) (1.10) Just as in the case of the equation of simple harmonic motion, the first two coefficients a0 and a1 in the power series can be determined from the initial conditions, dy y (0) = a0 , (0) = a1 . dx The recursion formula can be used to determine the remaining coefficients in the power series. Indeed it follows from (1.10) with n = 0 that a2 = − 2p a0 . 2·1 Similarly, it follows from (1.10) with n = 1 that a3 = 2 − 2p 2(p − 1) a1 = − a1 , 3·2 3! and with n = 2 that a4 = − 4 − 2p 2(2 − p) −2p 22 p(p − 2) a2 = a0 = a0 . 4·3 4·3 2 4! Continuing in this manner, we find that a5 = 6 − 2p 2(3 − p) 2(1 − p) 22 (p − 1)(p − 3) a3 = a1 = a1 , 5·4 5·4 3! 5! 11 8 − 2p 2(3 − p) 22 (p − 2)p 23 p(p − 2)(p − 4) a4 = a0 = − a0 , 6·5·2 6·5 4! 6! and so forth. Thus we find that a6 = y = a0 1 − 2p 2 22 p(p − 2) 4 23 p(p − 2)(p − 4) 6 x+ x− x + ··· 2! 4! 6! +...
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This document was uploaded on 01/12/2014.

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