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On a criterion for oscillatory solutions of a linear differential equation of the second order

Published online by Cambridge University Press:  24 October 2008

J. C. P. Miller
J. C. P. Miller
Affiliation:
The UniversityLiverpoor
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Extract

This paper gives a criterion for determining whether real, non-zero solutions of a linear differential equation of the second order have an infinite or a finite number of zeros, or, in short, are oscillatory or non-oscillatory, as the independent variable tends to infinity.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 36 , Issue 3 , July 1940 , pp. 283 - 287
Copyright
Copyright © Cambridge Philosophical Society 1940

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References

* Kneser, A.: (a) Math. Ann. 42 (1893), 409CrossRefGoogle Scholar; (b) Crelle's J. 117 (1897), 72.Google Scholar

† Fowler, R. H., Quart. J. Pure and App. Math. 45 (1914), 289.Google Scholar

‡ Cf. Ince, E. L., Ordinary differential equations (London, 1927), p. 215.Google Scholar

* Ince, E. L., loc. cit. p. 225. It is also shown, on p. 224, that if one solution of the equation (6) oscillates, so do all other solutions of the equation, and with the same rapidity.

† Negative zeros of (14) correspond to complex zeros of (6), but corresponding zeros are both real in the interval x 0 < x 1, if x 0 is sufficiently great.