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A stability result for the linear differential equation x”+f(t)x = 0

Published online by Cambridge University Press:  09 April 2009

K. W. Chang
K. W. Chang
Affiliation:
Department of Mathematics Research School of Physical Sciences Australian National UniversityCanberra
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Suppose that the real-valued function ƒ(t) is positive, continuous and monotonic increasing for t ≧ t0. If x = x (t) is a solution of the equation for for t ≧ t0, it is known that the solution x(t) oscillates infinitely often as t → ∞ and that the successive maxima of |x(t)| decrease, with increasing t. In particular x(t) is bounded as t → ∞.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 7 , Issue 1 , February 1967 , pp. 7 - 8
Copyright
Copyright © Australian Mathematical Society 1967

References

[1]Cesari, L., Asymptotic behaviour and stability problems in ordinary differential equations (Second edition, Springer-Verlag, Berlin 1963) (section 5.5).CrossRefGoogle Scholar
[2]Galbraith, A. S., McShane, E. J. and Parrish, G. B., ‘On the solutions of linear second-order differential equations’, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 247–249.CrossRefGoogle ScholarPubMed
[3]Hartman, P., ‘On oscillations with large frequencies’, Bull. Un. Mat. Ital. (3) 14 (1959), 62–65.Google Scholar
[4]Lazer, A. C., ‘A stability result for the differential equation y″+p(x)y = 0’, Michigan Math. J. 12 (1965), 193–196.CrossRefGoogle Scholar