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Asymptotic expansions in certain second order nonhomogeneous differential equations

Published online by Cambridge University Press:  26 February 2010

Thomas G. Hallam
Thomas G. Hallam
Affiliation:
The Florida State University, Tallahassee, Florida
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Extract

In the nonoscillation theory of ordinary differential equations the asymptotic behaviour of the solutions has often been described by exhibiting asymptotic expansions for these solutions. A fundamental illustration of this technique may be found in Hille [6] wherein the linearly independent solutions of a second order homogeneous differential equation were described by single termed asymptotic expansions. For the dominant solution, this result was successively extended by Waltman in [8] for a second order equation and in [9] for an nth order equation. A further generalization of these results appears in [3] where a complete nth order nonhomogeneous nonlinear differential equation was considered; again, asymptotic representations were given to describe the behaviour of the solutions of the differential equation. Moore and Nehari [7], Wong [10], [11], and Hale and Onuchic [2], also use asymptotic representations in discussing the behaviour of the solutions of certain differential equations. All of the above results are essentially perturbation problems with the unperturbed linear differential equation having the form y(n) = h(t) for some n and h(t).

Type
Research Article
Information
Mathematika , Volume 15 , Issue 1 , June 1968 , pp. 30 - 38
Copyright
Copyright © University College London 1968

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References

1.Coddington, E. A. and Levinson, N., Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955).Google Scholar
2.Hale, J. K. and Onuchic, N., “On the Asymptotic Behavior of Solutions of a Class of Differential Equations”, Contrbs. to Diff. Eqns. 2 (Wiley, New York, 1963), 61–75.Google Scholar
3.Hallam, T. G., “Asymptotic Behavior of the Solutions of an nth Order Nonhomogeneous Ordinary Differential Equation”, Trans. Amer. Math. Soc., 122 (1966), 177–194.Google Scholar
4.Hallam, T. G., “Asymptotic Behavior of the Solutions of a Nonhomogeneous Singular Equation”, J. Diff. Eqns., 3 (1967), 135–152.CrossRefGoogle Scholar
5.Hallam, T. G., “Asymptotic Expansions in a Nonhomogeneous Differential Equation”, Proc. Amer. Math. Soc, 18 (1967), 432–438.CrossRefGoogle Scholar
6.Hille, E., “Nonoscillation Theorems”, Trans. Amer. Math. Soc., 64 (1948), 234–252.CrossRefGoogle Scholar
7.Moore, R. A. and Nehari, Z., “Nonoscillation Theorems for a Class of Nonlinear Differential Equations”, Trans. Amer. Math. Soc., 93 (1959), 30–52.CrossRefGoogle Scholar
8.Waltman, P., “On the Asymptotic Behavior of a Nonlinear Equation”, Proc. Amer. Math. Soc. 15 (1964), 918–923.CrossRefGoogle Scholar
9.Waltman, P., “On the Asymptotic Behavior of the Solutions of an nth Order Equation”, Monatsch. Math., 69 (1965), 427–430.CrossRefGoogle Scholar
10.Wong, P. K., “Existence and Asymptotic Behavior of Proper Solutions of a Class of Second Order Nonlinear Differential Equations”, Pac J. Math., 13 (1963), 737–760.CrossRefGoogle Scholar
11.Wong, P. K., “On a Class of Nonlinear Fourth Order Differential Equations”, to appear.Google Scholar