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 this function.

First order differential equation.

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

Where f- some function of several variables.

Theorem of existence and uniqueness of a solution to a differential equation. Let in the differential equation (1.1) the function and its partial derivative be continuous on the open set G coordinate plane Ooh. Then:

1. For any point in the set G there will be 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 coincide for all those values ​​of the variable X, 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 X, g(y), N(y), Q(y)- variable functions u.

Differential equations with separable variables

To solve such an equation, it should be transformed to a form in which the differential and functions of the variable X will end up on one side of the equality, and the variable at- to 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 into the expression X

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

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

We rewrite solution (b) in the form x=± or x=C, Where C=±.

Incomplete differential equations

A first order differential equation (1.1) is called incomplete if the function f clearly depends on only one variable: either X, either from u.

There are two cases of such dependence.

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

it is easy to verify that its solution is the function

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

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

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

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

The book contains all educational material in accordance with the Ministry of Higher Education program for the course of differential equations for mechanical, mathematical and physics and mathematics specialties of universities. There is also a small amount of additional material related to technical applications. This allows you to select material for lectures depending on the profile of the university. The volume of the book is significantly reduced in comparison with existing textbooks by reducing additional material and selecting simpler proofs from those available in educational literature. The theory is presented in sufficient detail and is accessible not only to strong, but also to average students. Examples of solving typical problems are given with explanations. At the end of the paragraphs, the 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 issues presented are indicated, with references to the literature.

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Table of contents
Preface 5
Chapter 1 Differential equations and their solutions 7
§ 1. Concept of differential equation 7
§ 2. The simplest methods for finding solutions 14
§ 3. Methods for reducing 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 representation 27
§ 5. Existence and uniqueness of a solution 34
§ b. Continuation of solutions 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 any order 81
§ 11. Linear equations with constant coefficients 92
§ 12. Linear equations of second order 109
§ 13. Boundary value problems 115
§ 14. Linear systems with constant odds 124
§ 15. Exponential function of 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. Study of stability using Lyapunov functions 167
§ 20. Stability according to the first approximation 175
§ 21. Singular points 181
§ 22. Limit cycles 190
Chapter 5 Differentiability of a solution with respect to a parameter and its applications 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

Preface
The book contains a detailed presentation of all the issues of the course program on ordinary differential equations for mechanical-mathematical and physics-mathematical specialties 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 equations; material on stability theory has been expanded.
New material and some questions traditionally included in the course (for example, theorems on oscillating solutions), but not required for the first acquaintance with the theory of differential equations, are given in small print, the beginning and end of which are separated by horizontal arrows. Depending on the profile of the university and the areas of student training at the department, the choice remains 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 well-known textbooks for this course due to the reduction of additional (not included in the compulsory program) material and due to the selection of simpler proofs from those available in the educational literature.
The material is presented in detail and is accessible to students with an average level of training. Only classic ones are used
concepts mathematical analysis and basic information from linear algebra, including the Jordan form of the matrix. A minimum number of new definitions are introduced. After presenting the theoretical material, examples of its application are given with detailed explanations. The numbers of problems for exercises from the “Collection of problems on differential equations” by A. F. Filippov are indicated.
At the end of almost every paragraph, several directions in which research has developed on this issue, - directions that can be named using already known concepts, and for which there is literature in Russian.
Each chapter of the book has its own numbering of theorems, examples, and formulas. References to material from other chapters are rare and are given by indicating the chapter or paragraph number.

Filippov Aleksey 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 Ministry of Higher Education program for the course of differential equations for mechanical, mathematical and physics and mathematics specialties at universities. There is also a small amount of additional material related to technical applications. This allows you to select material for lectures depending on the profile of the university. The volume of the book is significantly reduced in comparison with existing textbooks by reducing additional material and selecting simpler proofs from those available in educational literature.
The theory is presented in sufficient detail and is accessible not only to strong, but also to average students. Examples of solving typical problems are given with explanations. At the end of the paragraphs, the 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 issues presented are indicated, with references to the literature (books in Russian).
Table of contents
Preface........................................................ .................5
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 reducing 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 a solution.................................34
§ b. Continued solutions.........................................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. .........1
§ 12. Linear equations of the second order.....................109
§ 13. Boundary value problems....................................115
§ 14. Linear systems with constant coefficients.....124
§ 15. Exponential function of a matrix................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. Study of stability using
Lyapunov functions........................167
§ 20. Stability according to the first approximation......175
§21. Singular points.........................181
§ 22. Limit cycles.........................190
Chapter 5
Differentiability of a solution with respect to a parameter and its applications.........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

Table of contents
Preface 5
Chapter 1 Differential equations and their solutions 7
§ 1. Concept of differential equation 7
§ 2. The simplest methods for finding solutions 14
§ 3. Methods for reducing 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 representation 27
§ 5. Existence and uniqueness of a solution 34
§ b. Continuation of solutions 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 second order 109
§ 13. Boundary value problems 115
§ 14. Linear systems with constant coefficients 124
§ 15. Exponential function of 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. Study of stability using Lyapunov functions 167
§ 20. Stability according to the first approximation 175
§ 21. Singular points 181
§ 22. Limit cycles 190
Chapter 5 Differentiability of a solution with respect to a parameter and its applications 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

The book contains all the educational material in accordance with the program of the Ministry of Higher Education on the course of differential equations for mechanical, mathematical and physics and mathematics specialties of universities. There is also a small amount of additional material related to technical applications. This allows you to select material for lectures depending on the profile of the university. The volume of the book is significantly reduced in comparison with existing textbooks by reducing additional material and selecting simpler proofs from those available in educational literature. The theory is presented in sufficient detail and is accessible not only to strong, but also to average students. Examples of solving typical problems 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. are indicated. Filippov and indicates some theoretical directions related to the issues presented, 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. By eliminating the unknowns directly from a given system, an equation with derivatives of a higher order is obtained, which is no easier to solve than the given system.

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

Table of contents
Preface 5
Chapter 1 Differential equations and their solutions 7
§ 1. Concept of differential equation 7
§ 2. The simplest methods for finding solutions 14
§ 3. Methods for reducing 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 representation 27
§ 5. Existence and uniqueness of a solution 34
§ b. Continuation of solutions 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 second order 109
§ 13. Boundary value problems 115
§ 14. Linear systems with constant coefficients 124
§ 15. Exponential function of 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. Study of stability using Lyapunov functions 167
§ 20. Stability according to the first approximation 175
§ 21. Singular points 181
§ 22. Limit cycles 190
Chapter 5 Differentiability of a solution with respect to a parameter and its applications 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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