Introduction to the theory of differential equations. Filippov A.F.

2nd ed., Rev. - M .: 2007. - 240 p.

The book contains all the educational material in accordance with the program of the Ministry of Higher Education for the course of differential equations for the mechanics and mathematics and physics and mathematics specialties of universities. There is also a small amount of additional material related to technical applications. This allows you to choose material for lectures, depending on the profile of the university. The volume of the book is significantly reduced in comparison with the existing textbooks due to the reduction of additional material and the selection of simpler proofs from those available in the educational literature. The theory is presented in sufficient detail and is accessible not only for strong, but also for average students. Examples of solving typical tasks are given with explanations. At the end of the paragraphs, numbers of problems for exercises from AF Filippov's "Collection of Problems on Differential Equations" are indicated, and some theoretical directions adjacent to the stated questions are indicated, with references to the literature.

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Table of contents
Foreword 5
Chapter 1 Differential Equations and Their Solutions 7
§ 1. The concept of a differential equation 7
§ 2. The simplest methods for finding solutions 14
§ 3. Methods for lowering the order of equations 22
Chapter 2 Existence and General Properties of Solutions 27
§ 4. Normal form of a system of differential equations and its vector notation 27
§ 5. Existence and uniqueness of a solution 34
§ b. Continuation of decisions 47
§ 7. Continuous dependence of the solution on the initial conditions and the right-hand side of equation 52
§ 8. Equations not resolved with respect to the derivative 57
Chapter 3 Linear Differential Equations and Systems 67
§ 9. Properties of linear systems 67
§ 10. Linear equations of any order 81
§ 11. Linear equations with constant coefficients 92
§ 12. Linear equations of the second order 109
§ 13. Boundary value problems 115
§ 14. Linear systems with constant coefficients 124
§ 15. The exponential function of the matrix J 137
§ 16. Linear systems with periodic coefficients 145
Chapter 4 Autonomous Systems and Resilience 151
§ 17. Autonomous systems 151
§ 18. The concept of stability 159
§ 19. Investigation of stability with the help of Lyapunov functions 167
§ 20. Stability in the first approximation 175
§ 21. Special points 181
§ 22. Limit cycles 190
Chapter 5 Differentiability of the solution by parameter and its application 196
§ 23. Differentiability of the solution with respect to the parameter 196
§ 24. Asymptotic methods for solving differential equations 202
§ 25. First integrals 212
§ 26. Partial differential equations of the first order 221
Literature 234
Index 237

Foreword
The book contains a detailed presentation of all the issues of the program of the course of ordinary differential equations for the mechanics and mathematics and physics and mathematics of universities, as well as some other issues relevant to the modern theory of differential equations and applications: boundary value problems, linear equations with periodic coefficients, asymptotic methods for solving differential equations; expanded material on the theory of stability.
New material and some questions traditionally included in the course (for example, theorems on oscillating solutions), but not necessary for the first acquaintance with the theory of differential equations, are given small print, the beginning and end of which are separated by horizontal arrows. Depending on the profile of the university and the areas of training students at the department, there is a choice of which of these questions to include in the course of lectures and the exam program.
The volume of the book is significantly less than the volume of known textbooks for this course due to the reduction of additional (not included in the compulsory curriculum) material and due to the selection of simpler proofs from those available in the educational literature.
The material is presented in detail and is available for students with an intermediate level of preparation. Only classic are used
concepts mathematical analysis and basic knowledge of linear algebra, including the Jordan form of a matrix. The minimum number of new definitions is introduced. After the presentation theoretical material examples of its application are given with detailed explanations. The numbers of problems for exercises from "Collection of problems on differential equations" by AF Filippov are indicated.
At the end of almost every section, several directions are listed in which research on this issue, - directions that can be named, using already known and, concepts, and for which there is literature in Russian.
Each chapter of the book has its own numbering of theorems, examples, formulas. References to the material of other chapters are rare and are given with the indication of the chapter or paragraph number.

Filippov Alexey Fedorovich Introduction to the theory of differential equations: Textbook. Ed. 2nd, rev. M., 2007 .-- 240 p.
The book contains all the educational material in accordance with the program of the Ministry of Higher Education for the course of differential equations for the mechanics and mathematics and physics and mathematics specialties of universities. There is also a small amount of additional material related to technical applications. This allows you to choose material for lectures, depending on the profile of the university. The volume of the book is significantly reduced in comparison with the existing textbooks due to the reduction of additional material and the selection of simpler proofs from those available in the educational literature.
The theory is presented in sufficient detail and is accessible not only for strong, but also for average students. Examples of solving typical tasks are given with explanations. At the end of the paragraphs, numbers of problems for exercises from A. F. Filippov's "Collection of Problems on Differential Equations" are indicated, and some theoretical directions related to the stated questions are indicated, with references to the literature (books in Russian).
Table of contents
Foreword ................................................. .................five
Chapter 1
Differential equations and their solutions ...................... 7
§ 1. The concept of a differential equation ...................... 7
§ 2. The simplest methods of finding solutions ........................ 14
§ 3. Methods for lowering the order of equations .................. 22
Chapter 2
Existence and general properties of solutions .......................... 27
§4. Normal view of a system of differential equations
and its vector notation .............................................. ..27
§ 5. Existence and uniqueness of the solution ...................... 34
§ b. Continuation of decisions .............................................. 47
§ 7. Continuous dependence of the solution on the initial conditions
and the right side of the equation .......................................... 52
§ 8. Equations not resolved with respect to the derivative ... 57
Chapter 3
Linear differential equations and systems ............ 67
§ 9. Properties of linear systems .......................................... 67
§ 10. Linear equations of any order ............................ 81

§ 11. Linear equations with constant coefficients. .........one
§ 12. Linear equations of the second order .............. 109
§ 13. Boundary value problems ............................ 115
§ 14. Linear systems with constant coefficients ..... 124
§ 15. The exponential function of the matrix ................ 137
§ 16. Linear systems with periodic coefficients ... 145
Chapter 4
Autonomous systems and stability ................. 151
§ 17. Autonomous systems ......................... 151
§ 18. The concept of sustainability ........................ 159
§ 19. Investigation of stability using
Lyapunov functions .......................... 167
§ 20. Stability in the first approximation ............. 175
§21. Special points ............................. 181
§ 22. Limit cycles .......................... 190
CHAPTER 5
Differentiability of the solution with respect to the parameter and its application ......... 196
§ 23. Differentiability of the solution with respect to the parameter ......... 196
§ 24. Asymptotic methods for solving differential
equations ............................... 202
§ 25. First integrals .......................... 212
§ 26. Partial differential equations of the first order ... 221
Literature .................................. 234
Index .......................... 237 Your browser does not seem to support iframes.

The book contains all educational material in accordance with the program of the Ministry of Higher Education on the course of differential equations for mechanics and mathematics and physics and mathematics specialties of universities. There is also a small amount additional material related to technical applications. This allows you to choose material for lectures, depending on the profile of the university. The volume of the book has been significantly reduced in comparison with the existing textbooks due to the reduction of additional material and the selection of simpler proofs from those available in educational literature... The theory is presented in sufficient detail and is accessible not only for strong, but also for average students. Examples of solving typical tasks are given with explanations. At the end of the paragraphs, the numbers of problems for exercises from the "Collection of problems on differential equations" by A.F. Filippov, and some theoretical directions are indicated that are adjacent to the stated questions, with references to the literature.

On the solution of nonlinear systems.
It is possible to find a solution using a finite number of actions only for some simple systems. When the unknowns are eliminated directly from the given system, an equation with derivatives greater than high order, which is no easier to solve than this system.

More often it is possible to solve the system by looking for integrable combinations. An integrable combination is either a combination of system equations containing only two variables
quantity and is a differential equation that can be solved, or such a combination, both sides of which are full differentials... From each integrable combination, the first integral of this system is obtained. When the unknowns are eliminated from the given system using the first integrals, the order of the derivatives does not increase.

Table of contents
Foreword 5
Chapter 1 Differential Equations and Their Solutions 7
§ 1. The concept of a differential equation 7
§ 2. The simplest methods for finding solutions 14
§ 3. Methods for lowering the order of equations 22
Chapter 2 Existence and general properties decisions 27
§ 4. Normal form of a system of differential equations and its vector notation 27
§ 5. Existence and uniqueness of a solution 34
§ b. Continuation of decisions 47
§ 7. Continuous dependence of the solution on initial conditions and the right side of equation 52
§ 8. Equations not resolved with respect to the derivative 57
Chapter 3 Linear Differential Equations and Systems 67
§ 9. Properties of linear systems 67
§ 10. Linear Equations any order 81
§ 11. Linear equations with constant coefficients 92
§ 12. Linear equations of the second order 109
§ 13. Boundary value problems 115
§ fourteen. Linear systems with constant coefficients 124
§ fifteen. Exponential function matrices J 137
§ 16. Linear systems with periodic coefficients 145
Chapter 4 Autonomous Systems and Resilience 151
§ 17. Autonomous systems 151
§ 18. The concept of stability 159
§ 19. Investigation of stability with the help of Lyapunov functions 167
§ 20. Stability in the first approximation 175
§ 21. Special points 181
§ 22. Limit cycles 190
Chapter 5 Differentiability of the solution by parameter and its application 196
§ 23. Differentiability of the solution with respect to the parameter 196
§ 24. Asymptotic methods for solving differential equations 202
§ 25. First integrals 212
§ 26. Partial differential equations of the first order 221
Literature 234
Subject Index 237.

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Introduction

Differential Equations.

A differential equation is an equation that connects the desired function of one or more variables, these variables and derivatives of various orders of the given function.

Differential equation of the first order.

Let us consider the questions of the theory of differential equations using the example of first-order equations solved with respect to the derivative, i.e. such that can be represented in the form

where f- some function of several variables.

Existence and uniqueness theorem for the solution of a differential equation. Suppose that in differential equation (1.1) the function and its partial derivative are continuous on the open set G coordinate plane Ohhu. Then:

1. For any point of the set G there is a solution y = y (x) equation (1.1) satisfying the condition y ();

2. If two solutions y = (x) and y = (x) equations (1.1) coincide for at least one value x =, i.e. if then these solutions are the same for all those values of the variable NS, for which they are defined. A first-order differential equation is called a separable equation if it can be represented as

or in the form

M (x) N (y) dx + P (x) Q (y) dy = 0,(1.3)

where, M (x), P (x)- some variable functions NS, g (y), N (y), Q (y)- variable functions at.

Separable Differential Equations

To solve such an equation, it should be transformed to the form in which the differential and functions of the variable NS will be in one part of the equality, and the variable at- in another. Then integrate both sides of the resulting equality. For example, from (1.2) it follows that = and =. Performing integration, we come to the solution of equation (1.2)

Example 1. Solve the equation dx = xydy.

Solution. Dividing the left and right sides of the equation by the expression NS

(at NS? 0), we arrive at equality. Integrating, we get

(since the integral on the left-hand side (a) is tabular, and the integral on the right-hand side can be found, for example, by replacing = t, 2ydy = 2tdt and .

Solution (b) can be rewritten as x = ± or x = C, where C = ±.

Incomplete Differential Equations

A first-order differential equation (1.1) is called incomplete if the function f explicitly depends on only one variable: either on NS, either from at.

There are two cases of such dependence.

1. Let the function f depend only on x. Rewriting this equation in the form

it is easy to verify that its solution is the function

2. Let the function f depend only on y, that is, equation (1.1) has the form

A differential equation of this kind is called autonomous. Such equations are often used in the practice of mathematical modeling and the study of natural and physical processes, when, for example, the independent variable NS plays the role of time, which is not included in the relationships that describe the laws of nature. In this case, the so-called equilibrium points, or stationary points - zeros of the function f(at), where the derivative y "= 0.

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