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On a problem of Littlewood concerning Riccati's equation

Published online by Cambridge University Press:  24 October 2008

H. P. F. Swinnerton-Dyer
Affiliation:
Trinity College, Cambridge

Extract

The study of the Riccati equation

plays an essential part in the ‘large parameter’ theory of the inhomogeneous van der Pol equation; see for example Littlewood(1), (2). The crucial result is Lemma B of (1), restated and proved as Lemma 5 of (2); for the present paper the relevant parts of it are as follows:

Lemma 1. Let z = z(x) be the solution of (1·1) which satisfies the initial condition z = 0 at x = 0, and assume α > 0. Then there is a unique β0 = β0(α) with the property that

(i) if β > β0 then z → − ∞ as x → + ∞;

(ii) if β < β0 then z → + ∞ at a vertical asymptote x = x0(α,β);

(iii) if β = β0 then z ≥ 0 in 0 ≤ x < + ∞ and z = x + β0 + o(1) as x → + ∞.

Moreover, β0(α) is a continuous monotone increasing function of α.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1969

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References

REFERENCES

(1)Littlewood, J. E.On non-linear differential equations of the second order, III. Acta Math. 97 (1957), 267308.CrossRefGoogle Scholar
(2)Littlewood, J. E.On non-linear differential equations of the second order, IV. Acta Math. 98 (1957), 1110.CrossRefGoogle Scholar
(3)Littlewood, J. E. Some problems in real and complex analysis. (In the Press.)Google Scholar
(4)Miller, J. C. P.The Airy integral (British Association Mathematical Tables, Part-Volume B). (Cambridge, 1946.)Google Scholar