A Course of Ordinary Differential Equations

Abstract

The textbook consists of five chapters. Chapter 1 deals with the basic concepts of the theory of ordinary differential equations. In addition, some special classes of first-order equations are studied in the chapter. Chapter 2 deals with linear differential equations and systems: the general theory of linear equations and systems with continuous coefficients; the theory of linear differential equations and systems with constant coefficients (in particular, functions of matrices are constructed and studied); introduction to the theory of boundaryvalue problem; methods of solving differential equations by using power series (in particular, the Bessel functions, the Bessel equations and their modified versions are considered). Chapter 3 deals with nonlinear systems: the existence and uniqueness theorem; the continuity and differentiability of solutions to the Cauchy problem with respect to parameters and initial conditions; the extension of solutions to the Cauchy problem; general integrals and methods of solving nonlinear systems, lineal and quasilinear partial differential equations. Chapter 4 deals with the Lyapunov stability: the Lyapunov methods, the stability of systems with constant coefficients, and the classification of equilibrium points for second-order systems. In Chapter 5 an introduction to the mathematical control theory are given. The concept of mapping (and operator) is a significant tool used in the textbook. Linear equations and systems are presented in an operator form. Then, properties of these operators are studied. Eventually, results for the equations and the systems based on the obtained properties are formulated. Moreover, all results for n-th order linear equations are obtained from the ones for n-th order linear systems by using a special mapping, except the results for n-th order linear equations with constant coefficients, which are studied directly. Within this approach, the existence and uniqueness theorem and the theorem of the continuity with respect to parameters and initial conditions are proved by using the concept of contractive mapping and the Banach fixed-point theorem. The method of successive approximations is also considered in the textbook to prove the existence and uniqueness theorem for linear systems with continuous coefficients. It is well-known that functions of matrices are a useful tool for obtaining solutions to linear differential systems. The method given in the textbook to construct functions of matrices is slightly nonstandard. The concept of function of matrices is considered for classes of functions, which are progressively enlarged, starting from polynomial function, then considering analytical function, and ending with functions defined on the spectrum of a matrix. As result, the concept of interpolation polynomial (including the Lagrange-Sylvester one) appears naturally, what is an important advantage of this approach. An introduction to the mathematical control theory is also an advantage of the textbook. This theory is intensively developed in recent years, and it has a number of applications. That is why the study of basic concepts of this theory is really important. The textbook is based on the author’s lectures on the subject at V. N. Karazin Kharkiv National University. It is certainly quite useful to undergraduate and postgraduate students studying the theory of ordinary differential equations and to engineers and scientists who are interested in this theory.