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Uniform approximate solutions of certain nth-order differential equations. I

Published online by Cambridge University Press:  24 October 2008

J. Heading
J. Heading
Affiliation:
University of Southampton
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Extract

Exact analytical solutions of certain second-order linear differential equations are often employed as approximate solutions of other second-order differential equations when the solutions of this latter equation cannot be expressed in terms of the standard transcendental functions. The classical exposition of this method has been given by Jeffreys (6); approximate solutions of the equation (using Jeffreys's notation)

are given in terms of solutions either of the equation

or of the equation

where h is a large parameter. A complete history of this technique is given in the author's recent text An introduction to phase-integral methods (Heading (5)).

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 59 , Issue 1 , January 1963 , pp. 95 - 110
Copyright
Copyright © Cambridge Philosophical Society 1963

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References

REFERENCES

(1)Heading, J.Proc. Cambridge Philos. Soc. 53 (1957), 399.CrossRefGoogle Scholar
(2)Heading, J.Proc. Cambridge Philos. Soc. 53 (1957), 419.CrossRefGoogle Scholar
(3)Heading, J.Proc. Cambridge Philos. Soc. 56 (1960), 329.CrossRefGoogle Scholar
(4)Heading, J.Quart. J. Mech. Appl. Math. 15 (1962), 215.CrossRefGoogle Scholar
(5)Heading, J.An introduction to phase-integral methods (Methuen; London, 1962).Google Scholar
(6)Jeffreys, H.Proc. Cambridge Philos. Soc. 49 (1953), 601.CrossRefGoogle Scholar