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Asymptotic solutions to linear differential equations with coefficients of the power order of growth, 1

Published online by Cambridge University Press:  14 November 2011

M. H. Lantsman
M. H. Lantsman
Affiliation:
Department of Mathematics, All-Union Correspondence Civil-Engineering Institute, Moscow 109029, U.S.S.R.
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Synopsis

We consider a method for determining the asymptotic solution to a sufficiently wide class of ordinary linear homogeneous differential equations in a sector of a complex plane or of a Riemann surface for large values of the independent variable z. The main restriction of the method is the condition that the coefficients in the equation should be analytic and single-valued functions in the sector for | z | ≫ 1 possessing the power order of growth for |z| → ∞. In particular, the coefficients can be any powerlogarithmic functions. The equation

can be taken as a model equation. Here ai are complex numbers, aij are real numbers (i = 1,2,…, n; j = 0, 1, …, m) and ln1 Z≡ln z, lnsz= lnlnS−1z = S = 2, … It has been shown that the calculation of asymptotic representations for solution to any equation in the class considered may be reduced to the solution of some algebraic equations with constant coefficients by means of a simple and regular procedure. This method of asymptotic integration may be considered as an extension (to equations with variable coefficients) of the well known integration method for linear differential equations with constant coefficients. In this paper, we consider the main case when the set of all roots of the characteristic polynomial possesses the property of asymptotic separability.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 100 , Issue 3-4 , 1985 , pp. 301 - 326
Copyright
Copyright © Royal Society of Edinburgh 1985

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References

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