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Chem Differential Eq HW Solutions Fall 2011 202

Chem Differential Eq HW Solutions Fall 2011 202 - A202...

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A202 Appendix A Ordinary Differential Equations: Review of Concepts and Methods We are in Case III. The solutions are of the form y 1 = m =0 a m x m - 1 and y 2 = ky 1 ln x + m =0 b m x m - 2 , with a 0 = 0, b 0 = 0. Let us determine y 1 . We use y instead of y to simplify the notation. We have y = m =0 a m x m - 1 ; y = m =0 ( m - 1) a m x m - 2 ; y = m =0 ( m - 1)( m - 2) a m x m - 3 . Plug into x 2 y + 4 xy + (2 - x 2 ) y = 0: m =0 ( m - 1)( m - 2) a m x m - 1 + m =0 4( m - 1) a m x m - 1 m =0 2 a m x m - 1 - m =0 a m x m +1 = 0 ( - 1)( - 2) a 0 x - 1 + m =2 ( m - 1)( m - 2) a m x m - 1 +4( - 1) a 0 x - 1 + m =2 4( m - 1) a m x m - 1 2 a 0 x - 1 + 2 a 1 + m =2 2 a m x m - 1 - m =2 a m - 2 x m - 1 = 0 2 a 1 + m =2 [( m - 1)( m - 2) a m + 4( m - 1) a m 2 a m - a m - 2 ] x m - 1 = 0 2 a
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