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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

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