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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Stability implications on the asymptotic behavior of second order differential equations

Author: John M. Bownds
Journal: Proc. Amer. Math. Soc. 39 (1973), 169-172
MSC: Primary 34D05
MathSciNet review: 0313596
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Abstract: Using some basic observations from stability theory, it is shown that the classical equation $ y'' + a(t)y = 0$ must have at least one solution $ y(t)$ such that $ \lim \sup (\vert y(t)\vert + \vert y'(t)\vert) > 0$ as $ t \to \infty $. The same conclusion holds for a nonlinear perturbation of this equation provided the linearization has a stable zero equilibrium. The results may be easily and naturally generalized to $ n$th order equations, although that generalization is not done here.

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Article copyright: © Copyright 1973 American Mathematical Society