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Chem Differential Eq HW Solutions Fall 2011 174

Chem Differential Eq HW Solutions Fall 2011 174 - A174...

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A174 Appendix A Ordinary Differential Equations: Review of Concepts and Methods Solutions to Exercises A.2 1. Equation: y - 4 y + 3 y = 0; Characteristic equation: λ 2 - 4 λ + 3 = 0 ( λ - 1)( λ - 3) = 0; Characteristic roots: λ 1 = 1; λ 2 = 3 General solution: y = c 1 e x + c 2 e 3 x 5 Equation: y + 2 y + y = 0; Characteristic equation: λ 2 + 2 λ + 1 = 0 ( λ + 1) 2 = 0; Characteristic roots: λ 1 = - 1 (double root) General solution: y = c 1 e - x + c 2 xe - x 9. Equation: y + y = 0; Characteristic equation: λ 2 + 1 = 0 Characteristic roots: λ 1 = i λ 2 = - i ; Case III: α = 0 , β = 1; General solution: y = c 1 cos x + c 2 sin x 13. Equation: y + 4 y + 5 y = 0; Characteristic equation: λ 2 + 4 λ + 5 = 0; Characteristic roots: λ = - 4 ± 16 - 20 2 = - 2 ± i ; λ 1 = - 2 + i, λ 2 = - 2 - i
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