Chem Differential Eq HW Solutions Fall 2011 184

Chem Differential - A184 Appendix A Ordinary Differential Equations Review of Concepts and Methods Wronskian cos x sin x − sin x cos x W x = = 1

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Unformatted text preview: A184 Appendix A Ordinary Differential Equations: Review of Concepts and Methods Wronskian: cos x sin x − sin x cos x W ( x) = = 1. We now apply the variation of parameters formula with g ( x) = yp sec x = 1 ; cos x −y2 g(x) dx + y2 W ( x) y1 g(x) dx W ( x) = y1 = − cos x = cos x · ln (| cos x|) + x sin x. sin x dx + sin x cos x dx Thus the general solution is y = c1 cos x + c2 sin x + cos x · ln (| cos x|) + x sin x. 29. x2y + 3 xy + y = indicial equation is √ x. The homogeneous equation is an Euler equation. The r2 + 2r + 1 = 0 ⇒ (r + 1)2 = 0. We have one double indicial root r = −1. Hence the solutions of the homogenous equation y1 = x−1 and y2 = x−1 ln x. Wronskian: 1 x W ( x) = −1 x2 1 x ln x 1−ln x x2 = 1 − ln x ln x 1 + 3 = 3. 3 x x x We now apply the variation of parameters formula with √ x 3 g ( x) = = x− 2 ; 2 x yp −y2 g(x) dx + y2 W ( x) = y1 = − = 1 − x = − = y1 g(x) dx W ( x) 1 2 2 3/2 4 1/2 =x. x x33 9 1 x ln x ln x 3 − 3 x x 2 dx + x x dv u ln x 1 3 −3 x x 2 dx x √ x dx + ln x x 1 2 3/2 x ln x − x3 1 x 2 dx 2 3/2 1 ln x 2 3/2 x dx + x 3 x x3 Thus the general solution is 4 y = c1 x−1 + c2x−1 ln x + x1/2. 9 33. x2y + 3xy + y = 0. See Exercise 29. 37. x2y + 7xy + 13 y = 0. Euler equation with α = 7, β = 13, indicial equation r2 + 6 r + 13 = 0; indicial roots: √ r = −3 ± −4; r1 = −3 − 2i, r2 = −3 + 2i. ...
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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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