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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
Affiliation:
The UniversityLiverpoor

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
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.