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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Asymptotic integration of a second order ordinary differential equation


Author: Jaromír Šimša
Journal: Proc. Amer. Math. Soc. 101 (1987), 96-100
MSC: Primary 34E10; Secondary 34C10
MathSciNet review: 897077
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Abstract: Equation (1) $ (r(t)x')' + f(t)x = 0$ is regarded as a perturbation of (2) $ (r(t)y')' + g(t)y = 0$, where the latter is nonoscillatory at infinity. It is shown that if a certain improper integral involving $ f - g$ converges sufficiently rapidly (but perhaps conditionally), then (1) has a solution which behaves for large $ t$ like a principal solution of (2). The proof of this result is presented in such a way that it also yields as a by-product an improvement on a recent related result of Trench.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0897077-X
PII: S 0002-9939(1987)0897077-X
Article copyright: © Copyright 1987 American Mathematical Society