Differential equations with separable variables. Filippov a. f. introduction to the theory of differential equations: textbook online Introduction to the theory of differential equations pdf

Introduction

Differential equations.

A differential equation is an equation that relates 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 solved with respect to the derivative, i.e. those that can be represented in the form

where f- some function of several variables.

Existence and uniqueness theorem for 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 Ohu. 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 X, for which they are defined. A first-order differential equation is called an equation with separable variables if it can be represented in the form

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 y.

Differential equations with separable variables

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

Example 1 solve the equation dx=xydy.

Decision. Dividing the left and right sides of the equation into the expression X

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

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

Solution (b) can be rewritten 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 explicitly depends on only one variable: either on X, either from y.

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

Filippov Alexey Fedorovich Introduction to the theory of differential equations: Textbook. Ed. 2nd, rev. M., 2007. - 240 p.
The book contains all educational material in accordance with the program of the Ministry of Higher Education in the course of differential equations for the mechanics and mathematics and physics and mathematics specialties of universities. There is also a small amount additional material associated with 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 available textbooks by reducing the additional material and choosing simpler proofs from those available in the textbooks.
The theory is presented in sufficient detail and is accessible not only for strong, but also for average students. Examples of solutions are given with explanations typical tasks. 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 are indicated that are adjacent to the questions outlined, with references to the literature (books in Russian).
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 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. Decisions continued..............................................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
§ ten. Linear equations any order.......................81

§ 11. Linear equations with constant coefficients. .........one
§ 12. Linear equations of the second order....................... 109
§ 13. Boundary Value Problems....................................115
§ fourteen. Linear systems with constant coefficients ..... 124
§ fifteen. Exponential function matrices...................137
§ 16. Linear systems with periodic coefficients... 145
Chapter 4
Autonomous Systems and Sustainability..................................151
§ 17. Autonomous systems .......................... 151
§ 18. The concept of stability .......................... 159
§ 19. Investigation of stability using
Lyapunov functions .........................167
§ 20. Stability in 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 a solution with respect to a parameter..........196
§ 24. Asymptotic methods for solving differential
Equations..............................202
§ 25. First integrals....................................212
§ 26. Equations with partial derivatives of the first order ... 221
Literature................................. 234
Index ...............................................237 Your browser does not seem to support iframes.

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 program of the Ministry of Higher Education in 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 available textbooks by reducing the additional material and choosing simpler proofs from those available in the textbooks. The theory is presented in sufficient detail and is accessible not only for strong, but also for 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 are indicated that are adjacent to the questions outlined, with references to the literature.

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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 for finding solutions 14
§ 3. Methods for lowering the order of equations 22
Chapter 2 Existence and general properties of solutions 27
§ 4. The 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 side of the equation 52
§ 8. Equations not solved 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 Sustainability 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. 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
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 physical and 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; 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 mandatory for a 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 student training, the department has a choice of which of these issues 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 mandatory program) material and due to the choice of simpler proofs from those available in the educational literature.
The material is presented in detail and accessible to students with an average level of preparation. Only classic ones are used.
concepts mathematical analysis and basic information from linear algebra, including the Jordan form of a matrix. The minimum number of new definitions is introduced. After the presentation of 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 are listed in which research on this issue has developed - 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 material in other chapters are rare and are given with the chapter or paragraph number.

The book contains all the educational material in accordance with the program of the Ministry of Higher Education in 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 available textbooks by reducing the additional material and choosing simpler proofs from those available in the textbooks. The theory is presented in sufficient detail and is accessible not only for strong, but also for 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. Filippov and indicate some theoretical directions related to the issues outlined, with references to the literature.

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

More often it is possible to solve the 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 parts of which are total differentials. From each integrable combination, the first integral of the given system is obtained. When the unknowns are excluded from the given system with the help of the first integrals, the order of the derivatives does not increase.

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