Unformatted text preview: + 1) + √ ∆ 2 , where the discriminant ∆ := ( λ + 1) 2 − 4( λ + 3). We consider two cases: i) If λ + 3 < 0, i.e. λ < − 3, then ∆ > ( λ + 1) 2 and the root r 2 > ( λ + 1) +  λ + 1  2 = 0 . Therefore, the solution x ( t ) = e r 2 t is unbounded as t → + ∞ . ii) If λ + 3 ≥ 0, i.e. λ ≥ − 3, then ∆ ≤ ( λ + 1) 2 . If ∆ < 0, then a fundamental solution set to (5.14) is e ( λ +1) t/ 2 cos ³ √ − ∆ t 2 ´ , e ( λ +1) t/ 2 sin ³ √ − ∆ t 2 ´ . (5.15) 272...
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 Spring '10
 Hald,OH
 Math, Differential Equations, Linear Algebra, Algebra, Equations, auxiliary equation, operator notation, ﬁrst equation, root r2

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