277_pdfsam_math 54 differential equation solutions odd

277_pdfsam_math 54 differential equation solutions odd -...

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Exercises 5.2 If ∆ 0, then < | λ +1 | and a fundamental solution set is { e r 1 t ,e r 2 t } , if ∆ > 0 , { e r 1 t ,te r 1 t } , if ∆ = 0 , (5.16) where both roots r 1 , r 2 are non-positive if and only if λ ≤− 1. For λ = 1w ehav e ∆=( 1+1) 2 4( 1+3) < 0, and we have a particular case of the fundamental solution set (5.15) (without exponential term) consisting of bounded functions. Finally, if λ< 1, then r 1 < 0, r 2 0, and all the functions listed in (5.15), (5.16) are bounded. Any solution x ( t ) is a linear combination of fundamental solutions and, therefore, all solutions x ( t ) are bounded if and only if 3 λ ≤− 1. 31. Solving this problem, we follow the arguments described in Section 5.1, page 242 of the text, i.e., x (
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