Chem Differential Eq HW Solutions Fall 2011 200

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ∞ ...
View Full Document

Ask a homework question - tutors are online