Chem Differential Eq HW Solutions Fall 2011 176

Chem Differential Eq HW Solutions Fall 2011 176 - A176...

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A176 Appendix A Ordinary Differential Equations: Review of Concepts and Methods Hence A = 1 / 8 , B = 3 / 32; y p = ( x 8 + 3 32 ) e - x ; and so the general solution y g = c 1 e x + c 2 e 3 x + ( x 8 + 3 32 ) e - x . 33. Equation: y + y = 1 2 + 1 2 cos 2 x ; Homogeneous equation: y + y = 0; Characteristic equation: λ 2 + 1 = 0 Characteristic roots: λ 1 = - i, λ 2 = i ; Solution of homogeneous equation: y h = c 1 cos x + c 2 sin x. To find a particular solution, we apply the method of undetermined coefficients. We also use our experience and simplify the solution by trying y p = 1 2 + A cos 2 x ; y p = - 2 A sin 2 x ; y p = - 4 A cos 2 x. Plug into the equation y + y = 1 2 + 1 2 cos 2 x : - 3 A cos 2 x + 1 2 = 1 2 + 1 2 cos 2
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