Chem Differential Eq HW Solutions Fall 2011 174

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

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Unformatted text preview: A174 Appendix A Ordinary Differential Equations: Review of Concepts and Methods Solutions to Exercises A.2 1. Equation: y −4y +3y = 0; λ −4λ +3 = 0 (λ − 1)(λ − 3) = 0; 2 Characteristic equation: Characteristic roots: λ1 = 1; General solution: y = c1ex + c2 e3x λ2 = 3 5 Equation: y +2y + y 0; λ +2λ +1 = 0 (λ + 1)2 Characteristic equation: Characteristic roots: = 0; 2 = λ1 = −1 (double root) y = c1 e−x + c2 x e−x General solution: 9. Equation: y +y = 0; Characteristic equation: λ2 + 1 =0 Characteristic roots: λ1 = i Case III: General solution: λ2 = −i; α = 0, β = 1; y = c1 cos x + c2 sin x 13. Equation: Characteristic equation: Characteristic roots: y +4y +5y = 0; λ +4λ +5 = √ −4 ± 16 − 20 λ= = −2 ± i; 2 λ1 = −2 + i, λ2 = −2 − i; 0; 2 Case III: General solution: α = −2, β = 1; y = c1e−2x cos x + c2 e−2x sin x 17. Equation: y −2y + y λ3 − 2λ2 + λ Characteristic equation: λ(λ − 1) Characteristic roots: General solution: λ1 = 0; 2 = 0; =0 = 0; λ2 = 1 (double root) y = c1 + c2 ex + c3 x ex 21. Equation: Characteristic roots: General solution: = 0; λ3 − 3λ2 + 3λ − 1 =0 (λ − 1)3 Characteristic equation: y − 3 y + 3y − y = 0; λ1 = 1 (multiplicity 3); y = c1 ex + c2 x ex + c3 x2ex ...
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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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