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Chem Differential Eq HW Solutions Fall 2011 192

Chem Differential Eq HW Solutions Fall 2011 192 - A192...

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A192 Appendix A Ordinary Differential Equations: Review of Concepts and Methods 5. For the differential equation y - y = 0, p ( x ) = 0 is its own power series expansion about a = 0. So a = 0 is an ordinary point. To solve, 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 . Then y - y = m =2 m ( m - 1) a m x m - 2 - m =0 a m x m = m =0 ( m + 2)( m + 1) a m +2 x m - m =0 a m x m = m =0 [( m + 2)( m + 1) a m +2 - a m ] x m . So y - y = 0 implies that ( m + 2)( m + 1) a m +2 - a m = 0 a m +2 = a m ( m + 2)( m + 1) for all m 0 . So a 0 and a 1 are arbitrary; a 2 = a 0 2 , a 4 = a 2 4 · 3 = a 0 4! , a 6 = a 4 6 · 5 = a 0 6! , . . . a 2 n = a 0 (2 n )! . . Similarly, a 2 n +1 = a 1 (2 n + 1)! , and so y = a 0 n =0 1 (2 n )! x 2 n + a 1 n =0 1 (2 n + 1)! x 2 n +1 = a 0 cosh x + a 1 sinh x. 9. For the differential equation y +2 xy + y = 0, p ( x ) = 2 x is its own power series
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