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Approximate Solution of Linear Second Order Differential Equations

Published online by Cambridge University Press:  04 July 2016

William Squire
William Squire*
Affiliation:
Bell Aircraft Corporation, Buffalo, New York.
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Extract

In a recent paper(1) Collar discussed some aeronautical applications of linear differential equations with variable coefficients. Part of the paper deals with the approximate solution:

of the differential equation

This asymptotic approximation, which dates back at least to Liouville, has an interesting history(2). It is widely known as the WKB approximation because of its use in quantum theory by Wentzel, Kramers and Brillouin. It has been applied to compressible flow by Imai(3).

While very useful it breaks down at the zeros of n(t) and there are problems in joining solutions passing through such points. Recently(2,4,5) extensions of the approximation which circumvent this difficulty have been developed. This note deals with the extension due to Bailey.

This approximation can be developed from the equivalence of equation (2) and the Riccati equation: —

Type
Research Article
Information
The Aeronautical Journal , Volume 63 , Issue 582 , June 1959 , pp. 368 - 369
Copyright
Copyright © Royal Aeronautical Society 1959

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References

1.Collar, A. R. (1957). On the Stability of Accelerated Motion: Some Thoughts on Linear Differential Equations with Variable Coefficients. The Aeronautical Quarterly, Vol. VIII, p. 309, 1957.Google Scholar
2.Bailey, V. A. (1954). Reflection of Waves by an Inhomogeneous Medium. Physical Review, Vol. 76, p. 865, 1954.CrossRefGoogle Scholar
3.Imai, I. (1949-50). Application of the W.K.B. Method to the Flow of a Compressible Fluid. Journal of Mathematics and Physics, 28, 173-182, 205–214, 1949-50.Google Scholar
4.Dingle, R. B. (1956). The Method of Comparison Equations in the Solution of Linear Second-Order Differential Equations (Generalised W.K.B. Method). Applied Scientific Research Series B, Vol. 5, p. 345. 1956.Google Scholar
5.Hecht, C. E. and Mayer, J. E. (1957). Extension of the W.K.B. Equation. Physical Review, Vol. 106, p. 1156, 1957.CrossRefGoogle Scholar