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Differential Equations, Vol. 41, No. 6, 2005, pp. 797–807. Translated from Differentsial'nye Uravneniya, Vol. 41, No. 6, 2005, pp. 761–770. Original Russian Text Copyright c 2005 by Levenshtam. ORDINARY DIFFERENTIAL EQUATIONS Asymptotic Integration of Di erential Equations with Oscillatory Terms of Large Amplitudes: I V. B. Levenshtam Rostov State University, Rostov, Russia Received May 19, 2004 The main theorems of the classical theory of the Bogolyubov averaging method [1] deal with systems of ordinary di erential equations in the so-called standard form, i.e., representable in the form dx = f (x;!t); (1) dt where ! is a large parameter and f (x; ) is a vector function that has an average with respect to : f (x) = lim f (x; )d hfi(x): T!1 T In numerous subsequent studies (see the extensive bibliography in [2]), many results of the classical theory were generalized to various new classes of equations. In the theory of ordinary di erential equations, a large class including equations in standard form and well-known systems with rapidly rotating phase [3] is formed by systems of equations with rapid and slow variables [4, 5], which can be represented in the form dx dy = f (x;y;!t);
Differential Equations – Springer Journals
Published: Jul 27, 2005
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