Chem Differential Eq HW Solutions Fall 2011 189

Chem Differential Eq HW Solutions Fall 2011 189 - ± m =0 a...

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Section A.4 The Power Series Method, Part I A189 25. Let a be any real number ± = 0, then 1 x = 1 a - a + x = 1 a · 1 1 - ( a - x a ) = 1 a ± n =0 ² a - x a ³ n = 1 a ± n =0 ( - 1) n ( x - a ) n a n . The series converges if ´ ´ ´ ´ a - x a ´ ´ ´ ´ < 1o r | a - x | < | a | . 29. Recall that changing m to m - 1 in the terms of the series requires shifting the index of summation up by 1. This is what we will do in the second series: ± m =1 x m m - 2 ± m =0 mx m +1 = ± m =1 x m m - 2 ± m =1 ( m - 1) x m = ± m =1 x m µ 1 m - 2( m - 1) = ± m =1 - 2 m 2 +2 m +1 m x m . 33. Let y =
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Unformatted text preview: ± m =0 a m x m y ± = ∞ ± m =1 ma m x m-1 . Then y ± + y = ∞ ± m =1 ma m x m-1 + ∞ ± m =0 a m x m = ∞ ± m =0 ( m + 1) a m +1 x m + ∞ ± m =0 a m x m = ∞ ± m =0 [( m + 1) a m +1 + a m ] x m 37. Let y = ∞ ± m =0 a m x m y ± = ∞ ± m =1 ma m x m-1 y ±± = ∞ ± m =2 m ( m-1) a m x m-2 ....
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This note was uploaded on 12/22/2011 for the course MAP 3305 taught by Professor Stuartchalk during the Fall '11 term at UNF.

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