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An extension of the Liouville–Green asymptotic formula for oscillatory second-order differential equations

Published online by Cambridge University Press:  14 November 2011

J. S. Cassell
J. S. Cassell
Affiliation:
Faculty of Computing, Management Science, Mathematics and Statistics, City of London Polytechnic, 31 Jewry Street, LondonEC3N 2EY
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Synopsis

Conditions, generalizing the usual Liouville–Green conditions, are obtained under which the equation {r(x)y′}′ + q(x)y = 0, where q and r are positive and have derivatives of a sufficiently high order, has solutions of the form y ∼ (rq)–¼e10. 0 ∼ f (q/r)½ dx as x →∞. A related result is obtained for a system of two first-order equations y′ = A(x)y, where the eigenvalues of A are pure imaginary.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 100 , Issue 3-4 , 1985 , pp. 181 - 190
Copyright
Copyright © Royal Society of Edinburgh 1985

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References

1Atkinson, F. V.. Asymptotic formulae for linear oscillations. Proc. Glasgow Math. Assoc. 3 (1957), 105–111.CrossRefGoogle Scholar
2Atkinson, F. V., Eastham, M. S. P. and McLeod, J. B.. The limit-point, limit-circle nature of rapidly oscillating potentials. Proc. Roy. Soc. Edinburgh Sect. A 76 (1977), 183–196.CrossRefGoogle Scholar
3Coppel, W. A.. Stability and asymptotic behaviour of differential equations (Boston: Heath, 1965).Google Scholar
4Eastham, M. S. P.. The Liouville–Green asymptotic theory for second-order differential equations: a new approach and some extensions. Ordinary differential equations and operators. Lecture Notes in Mathematics 1032 (Berlin: Springer, 1983).Google Scholar
5Eastham, M. S. P.. Asymptotic formulae of Liouville–Green type for higher order differential equations. J. London Math. Soc. 28 (1983), 507–518.CrossRefGoogle Scholar
6Hartman, P. and Wintner, A.. Asymptotic integration of linear differential equations. Amer. J. Math. 77 (1955), 45–86.CrossRefGoogle Scholar