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Chem Differential Eq HW Solutions Fall 2011 186

Chem Differential Eq HW Solutions Fall 2011 186 - A186...

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A186 Appendix A Ordinary Differential Equations: Review of Concepts and Methods which implies (3). 49. 3 y + 13 y + 10 y = sin x, y 1 = e - x . As in the previous exercise, let y 1 = e - x , y = ve - x , y = ve - x - ve - x , y = v e - x - 2 ve - x + ve - x . Then 3 y + 13 y + 10 y = sin x 3( v e - x - 2 ve - x + ve - x ) +13( ve - x - ve - x ) + 10 ve - x = sin x 3 v + 7 v = e x sin x v + 7 3 v = 1 3 e x sin x. We now solve the first order o.d.e. in v : e 7 x/ 3 v + 7 3 e 7 x/ 3 v = e 7 x/ 3 1 3 e x sin x d dx e 7 x/ 3 v = 1 3 e 10 x/ 3 sin x e 7 x/ 3 v = 1 3 1 3 e 10 x/ 3 sin xdx = 1 3 e 10 x/ 3 ( 10 3 ) 2 + 1 ( 10 3 sin x - cos) + C v = e x 109 (10 sin x - 9 3 cos) + C. (We used the table of integrals to evaluate the preceding integral. We will use it
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