Tutorial9

Tutorial9 - Four sequences of coefficients, starting with a...

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Unformatted text preview: Four sequences of coefficients, starting with a ,a 1 ,a 2 and a 3 decouple. In particular the sequences are a ,a 4 ,a 8 ,a 12 ,... ; a 1 ,a 5 ,a 9 ,a 13 ,... ; a 2 ,a 6 ,a 10 ,a 14 ,... ; a 3 ,a 7 ,a 11 ,a 15 ,.... Since a 2 = a 3 = 0 , by recurrence relation the last two sequences vanish identically. Starting with a , we have a , a 4 =- k 2 3 · 4 a , a 8 = k 4 3 · 4 · 7 · 8 a , a 12 =- k 6 3 · 4 · 7 · 8 · 11 · 12 a ; and starting with a 1 , a 1 , a 5 =- k 2 4 · 5 a 1 , a 9 = k 4 4 · 5 · 8 · 9 a 1 , a 13 =- k 6 4 · 5 · 8 · 9 · 12 · 13 a 1 . Finally the general solution can therefore written as y ( x ) = a y ( x ) + a 1 y 1 ( x ) , where y ( x ) = 1- k 2 x 4 3 · 4 + k 4 x 8 3 · 4 · 7 · 8- k 6 x 12 3 · 4 · 7 · 8 · 11 · 12 + ··· = 1 + ∞ X m =0 (- 1) m +1 ( k 2 x 4 ) m +1 3 · 4 · 7 · 8 ··· (4 m + 3)(4 m + 4) , y 1 ( x ) = x- k 2 x 5 4 · 5 + k 4 x 9 4 · 5 · 8 · 9- k 6 x 13 4 · 5 · 8 · 9 · 12 · 13 + ··· = x " 1 + ∞ X m =0 (- 1) m +1 ( k 2 x 4 ) m +1 4 · 5 · 8 · 9 ··· (4 m...
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This note was uploaded on 01/28/2011 for the course MATH 150 taught by Professor T.qian during the Spring '09 term at HKUST.

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Tutorial9 - Four sequences of coefficients, starting with a...

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