# Asymptotic Methods for Ordinary Differential Equations by R. P. Kuzmina (auth.)

By R. P. Kuzmina (auth.)

In this booklet we think about a Cauchy challenge for a approach of normal differential equations with a small parameter. The booklet is split into th ree elements in keeping with 3 ways of concerning the small parameter within the process. partially 1 we research the quasiregular Cauchy challenge. Th at is, an issue with the singularity integrated in a bounded functionality j , which is dependent upon time and a small parameter. This challenge is a generalization of the regu larly perturbed Cauchy challenge studied by means of Poincare [35]. a few differential equations that are solved by way of the averaging process might be diminished to a quasiregular Cauchy challenge. for example, in bankruptcy 2 we think about the van der Pol challenge. partly 2 we research the Tikhonov challenge. this can be, a Cauchy challenge for a method of standard differential equations the place the coefficients via the derivatives are integer levels of a small parameter.

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**Sample text**

8) and the inequalities IgEI ~ te + lalE 2 ~ C < 1. 3. 9) where a is a constant independent of x, t, E. L) = O. 10) SOLUTION EXPA NSIO NS OF THE QUASIREG ULAR CAUCHY. . 10) takes the form 00 z = (e h - 1) e + L eh(e h - l )k-l c k, h == t + aJ-L sin ( ~ ) . 10) for t he pro blem (8. 9) is eq ual to x = (e9 - 1) e + 00 L g==t+ac sin(~). 8. 11) . The variational eq uation t akes t he form ~; = [1 + a cos (~ )] (. 4) for U a re not valid . 11). 9) exists , is unique , a nd sat isfies t he inequ ali t y for 0 ~ t ~ T , 0

2. 5) were a is a constant independent of x, t, c . Take f(t, c) = cos(t/c) and consider the problem with two small parameters ~: = [1 + a cos ( ;) ] (z + c) 2 , Z ( 0, c, It) = O. 5) is equal to 00 x= L gk -lck , g == t + ae sin (~) . 8. 7) . The var iational equation takes t he form d( - =0. dt The Cauchy mat rix is equal to U(t , 5 , It) = 1. 3) for U is not valid . 7) . 3 , so we do not consider it here . 5) exists, is unique, and satisfies the ineq ua lity 46 CHAPTER 1 for 0 ~ t ~ T, 0

Let J be an integer , 1 ~ J ~ N; x be a vector of J components of the vector x , D be a set in vector space RN+2 3 (x, t,c). By D* denote the set D* {(x, t , c): Ilxll ~ 8, t ~ 0, ~ e ~ E}.