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Integrals of the Differential Equations ẍ + f ( s ) x = 0

Published online by Cambridge University Press:  20 November 2018

R. Datko
R. Datko*
Affiliation:
McGill University
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In this note we consider a relatively ancient stability problem: the behaviour of solutions of the second order differential equation ẍ + f(s) x = 0, where f(s) tends to plus infinity as s tends to plus infinity. An extensive survey of the literature concerning this problem and a resume of results may be found in [ l ]. More recently McShane et a l. [2] have shown that the additional assumption f(s) ≥ 0 is not sufficient to guarantee that all solutions tend to zero as s tends to infinity. Our aim is to demonstrate a new criterion for which all solutions do have the above property. This criterion overlaps many of the cases heretofore considered.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 10 , Issue 2 , June 1967 , pp. 191 - 196
Copyright
Copyright © Canadian Mathematical Society 1967

References

1. Cesari, L., Asymptotic Behaviour and Stability Problems in Ordinary Differential Equations. Ergebnisse der Mathematik and Ihrer Grenzgebiete. NF 16 (1966) New York.Google Scholar
2. Galbraith, A. S., McShane, E. J. and Parrish, G. B., On the solutions of linear second order equations. Proc. Nat. Acad. Scie. U.S.A. 53 (1966), pages 247-249.Google Scholar
3. Biernacki, M., Sur l′equation x + A(t) Ẍ = 0. Prace Mat. F17 40 (1933), pages 163-171.Google Scholar