Chem Differential Eq HW Solutions Fall 2011 200

# Chem Differential Eq HW Solutions Fall 2011 200 - A200...

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Unformatted text preview: A200 Appendix A Ordinary Diﬀerential Equations: Review of Concepts and Methods Plug into x y + (1 − x) y + y = 0: ∞ ∞ m(m − 1)am xm−1 + m=1 mam xm−1 m=1 ∞ ∞ mam xm + − m=1 am xm =0 m=0 ∞ ∞ (m + 1)mam+1 xm + m=0 (m + 1)am+1 xm m=0 ∞ ∞ mam xm + − m=1 ∞ am xm =0 m=0 ∞ (m + 1)mam+1 xm + a1 + m=1 (m + 1)am+1 xm m=1 ∞ ∞ mam xm + a0 + am xm =0 [(m + 1)mam+1 + (m + 1)am+1 − mam + am ] xm =0 − m=1 m=1 ∞ a0 + a1 + m=1 ∞ (m + 1)2 am+1 + (1 − m)am xm a0 + a1 + =0 m=1 This gives a0 + a1 = 0 and the recurrence relation: For all m ≥ 1, am+1 = − 1−m am . (m + 1)2 Take a0 = 1. Then a1 = −1 and a2 = a3 = · · · = 0. So y1 = 1 − x. We now turn to the second solution: (use y1 = 1 − x, y1 = −1, y1 = 0) ∞ y = bm xm ; y1 ln x + m=0 ∞ y = y1 ln x + y1 mbm xm−1; + x m=0 y = y1 ln x + 2 1 m(m − 1)bm xm−2. y − y1 + x 1 x2 m=0 ∞ ...
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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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