I. Overview of the course

(1) Course hours and credits

Course code: 1302, major: Mathematics and Applied Mathematics (Teachers College), the sixth semester starts, with a total course time of 72 hours and 4 credits. The total course hours include 54 hours and 18 hours of exercises.

(b) the nature of the course

Ordinary differential equations uses the basic knowledge of mathematical analysis, advanced algebra and complex variable function to introduce the common solutions and basic theories of ordinary differential equations. It will serve the students of mathematics, mechanics and physics in their future study, and it is an indispensable basic training for integrating mathematics with practice and flexibly using various mathematical methods. It is a compulsory course for university majors.

(3) Teaching purpose

Ordinary differential equation? It is a main compulsory course for undergraduate education majors in normal universities after basic courses such as mathematical analysis and advanced algebra. This paper mainly introduces the common solutions and basic theories of ordinary differential equations to students, including the solutions of several common first-order differential equations and higher-order linear differential equations and equations, as well as the most important theoretical basis in differential equation theory: the existence and uniqueness theorem of solutions and the continuation theorem of solutions. While improving students' ability to solve practical problems, this paper briefly introduces the basic ideas and methods of this course, and cultivates students' ability to analyze general differential equations.

(D) the links and division of labor between this course and other courses

The course Ordinary Differential Equations is based on mathematical analysis, advanced algebra, complex variable function and other knowledge, which is helpful to the students of mathematics department in their future study. Such as: differential geometry, partial differential equations and other courses. Ordinary differential equation is a course closely linked with practice, and it is an indispensable basic training for integrating mathematics with practice and flexibly using various mathematical methods.

Second, the basic content and requirements of course teaching

(A) Teaching requirements

1, we should study ordinary differential equations in combination with students' actual level and ability.

2, master the basic method of solving differential equations (separation of variables method; Appropriate equation; First order linear equation; Homogeneous equation, Bernoulli equation, Riccati equation and integral factor method in elementary transformation method)

3. Master the most basic theoretical basis: the existence and uniqueness theorem of the solution and the continuation theorem of the solution.

4. Master some theorems and basic solutions of higher-order linear differential equations and linear differential equations.

5. Understand the concepts of odd solution and envelope and their solutions.

(b) Total class hours and class hour allocation

1, total class hours: 18 4=72 (class hours)

2, course hours allocation table

Chapter content hours

Chapter 1 Basic Concepts 4

Chapter II Elementary Integral Method 20

Chapter III Picard Theorem 12

The fourth chapter strange solution 8

Chapter 5 Higher-order Differential Equation 8

Chapter VI System of Higher Order Differential Equations 20

A total of 72 people

(3) Teaching content

The first chapter is thread theory.

(A) teaching objectives and requirements

Master the definition of differential equations and solutions, master the geometric interpretation of differential equations and solutions and the basic methods of linear element field.

(B) Teaching focus and difficulties

1, the definition and solution of differential equation

2, the basic practice of line element field

(3) Teaching methods

Give priority to lectures, give more examples and do more exercises.

(D) Teaching content

Section 1 Definition of Differential Equations and Solutions

1, the concept of ordinary differential equation.

2. The concept of solutions of ordinary differential equations.

(1) general solution (general integral)

(2) Special solutions

3. Initial value problem (Cauchy problem).

4. How to find a differential equation satisfied by a curve family?

In the second quarter, differential equations and geometric explanations of their solutions

1, the concepts of integral curve, line element, line element field and direction field.

2. Geometric interpretation of differential equations and their solutions.

3. How to make linear fields of some simple differential equations?

Chapter II Elementary Integration Method

(A) teaching objectives and requirements

Mastering several types of elementary integration methods skillfully can quickly determine the types of equations and then solve them. Mastering these methods and skills is the basic training for learning this course and other branches.

(B) Teaching focus and difficulties

1, the judgment condition of the appropriate equation and how to solve the appropriate equation with the formula.

2. Solve the variable separable equation.

3. The formal characteristics, solving formulas and five-point properties of first-order linear differential equations.

4. Elementary transformation method.

5. Integral factor method.

6. How to find the equiangular trajectory family and orthogonal trajectory family of the known curve family?

(3) Teaching methods

Give priority to classroom teaching and give more examples. Assign a certain amount of exercises as homework after class, and urge students to learn by correcting homework and quizzes in class, so as to improve their attention to the contents of this chapter.

(D) Teaching content

The first section of the appropriate equation

1, definition of appropriate equation (fully differential equation).

2. Theorem: How to judge a symmetric equation as an appropriate equation and solve it with a formula.

In the second quarter, equations with separable variables

1, the definition of variable separable equation.

2. Supplement: Some lost solutions need to be found back.

3. Direct integral solution.

4. Exercise 4 after class (involving physical knowledge).

In the third quarter, first-order linear equations

1, the definition of first-order linear equation.

2. The solution formula of first-order homogeneous linear equations.

3. The solution formula of first-order nonhomogeneous linear equations.

4. Understand the constant variation method.

5. Five properties of the solution of the first-order linear equation and some exercises.

Section 4 Elementary Transformation Method

1. Give two examples to show that some insoluble equations can be solved by elementary transformation.

2, homogeneous equation:, do transformation to solve.

3. How to transform the shape to solve the equation?

4. bernhard equation has been transformed.

5. The simplest nonlinear equation in the form of Riccati equation is only a general understanding.

Section 5 Integral Factor Method

1, the definition of the integral factor.

2. Theorems 3 and 4 give the integral factors of two kinds of special differential equations respectively.

3. Further introduce integration factors through grouping.

Theorem 6 is the basis of some exercises after class and will be explained in detail.

Section 6 Application Examples

1, the definitions of equiangular trajectory family and orthogonal trajectory family.

2. How to find the equiangular locus family and orthogonal locus family of the known curve family?

Chapter III Picard Theorem

(A) teaching objectives and requirements

The existence and uniqueness theorem, also known as Picard theorem, is a basic theorem in the theory of differential equations. It is necessary to remember and deeply understand the contents of Picard's theorem. With regard to the idea and method of proof, the Picard sequence is constructed by successive iteration method, and it needs to be mastered to prove that Picard sequence converges uniformly to the unique solution of the equation. The continuation theorem of the solution is to discuss the existence interval of some solutions of the equation, select part of the teaching materials of Sun Yat-sen University and Northeast Normal University, and supplement some examples and exercises after class. Inference is mainly used to generalize theorems.

(B) Teaching focus and difficulties

1, piccard theorem: Lipschitz condition, substitution condition.

2. Proof process.

3. Supplementary explanation.

4. Extension theorem and inference of solution.

5. Several important typical examples.

(3) Teaching methods

Classroom teaching is the main and meticulous. Students are required to review carefully after class and complete supplementary homework.

(D) Teaching content

The first section Picard theorem

1. Introduce the concept of Lee's condition and its substitution condition: continuous partial derivative of pair.

2. The content and proof process of Picard's theorem.

Section II Extension of Solutions

1, introducing Lipschitz condition.

2. Give an example to illustrate the extension of the solution.

3. Introduce the inference of continuation theorem and solution.

4. Use theorems and inferences to do supplementary exercises and arrange supplementary homework questions.

Chapter 4 Singular Solutions

(A) teaching objectives and requirements

Master two methods to solve the first-order implicit differential equation: differential method and parameter method. Master the concept of singular solution and the method of finding singular solution. Master the concept of envelope and its solution. Master the types and solutions of Clero equation.

(B) Teaching focus and difficulties

1, the key to solving the first-order implicit differential equation is to master the differential method and parameter method.

2. The concept of singular solution and the method of finding singular solution. These two theorems should be freely applied.

3. The concept of envelope and the solution of envelope should be applied freely.

4. The relationship between singular solution and envelope.

5. Types and solutions of Clero equation.

(3) Teaching methods

Give priority to classroom teaching, supplemented by examples and exercises, broaden your horizons and do more exercises.

(D) Teaching content

Section 1 First-order Implicit Differential Equation

1, and the formal equation is solved by differential method.

2. The shape equation is solved by parameter method.

3. The concept of envelope and how to find the envelope of curve family with two theorems.

4. The concept of singular solution and how to use two theorems to find the singular solution of the equation.

5. Types and solutions of Clero equation.

Chapter V Higher-order Differential Equations

(A) teaching objectives and requirements

For several special types of higher-order differential equations that can be reduced in order, we must master their solutions; Can realize the mutual transformation between order differential equation and order differential equation, and find out the relationship between their solutions; Two vector forms of famous standard differential equations and uniqueness of solutions to initial value problems.

(B) Teaching focus and difficulties

1, solutions of several special types of reducible higher order differential equations.

( 1)

(2)

(3) Inherent derivative equation.

2. By introducing variables, order differential equations and standard differential equations are transformed into each other.

3. Vector form of standard differential equation of order.

(3) Teaching methods

Give priority to classroom teaching, supplemented by students doing exercises after class.

(D) Teaching content

Section 1 Several Special Types of Decreasable Higher-order Differential Equations

1, the solution of the shape equation, make.

2. The solution of the shape equation makes.

3. Appropriate derivative equation.

4. supplementary exercises.

Differential equations in linear space in the second quarter

1, reciprocity between order differential equation and standard differential equation, introducing variables.

2. Two vector forms of second-order standard differential equations are commonly used.

3. Existence and uniqueness of solutions to initial value problems of standard differential equations of order.

Chapter VI Linear Differential Equations

(A) teaching objectives and requirements

Mastering the general theory of linear differential equations is mainly to understand the algebraic structure of all its solutions, and the central problem is the basic solution matrix of homogeneous linear differential equations. Any solution of nonhomogeneous equation can be obtained by integrating the basic solution matrix. For linear differential equations with constant coefficients, the basic solution matrix should be obtained by finding the characteristic roots. For higher order linear differential equations, it is required to find the fundamental system of solutions of homogeneous equations, and then find the general solution. For two special forms of the right-hand side of non-homogeneous equation, we can find the corresponding special solution, and then find the general solution.

(B) Teaching focus and difficulties

1 and the construction of general solutions of homogeneous linear differential equations.

2. Liouville formula.

3. Two properties of the basic solution matrix.

4. The general solution construction and solving formula of nonhomogeneous equation.

5. Basic solution matrix of linear differential equations with constant coefficients.

(1) is obtained by Jordan canonical form.

(2) Obtained by the method of undetermined exponential function.

6. General solution formula of non-homogeneous linear differential equation with constant coefficients.

7. Some equation theories are extended to higher order linear differential equations.

8. The construction of general solution of high-order homogeneous linear differential equation.

9. Construction of general solutions of two kinds of higher-order nonhomogeneous linear differential equations.

(3) Teaching methods

Give priority to classroom teaching, and deepen students' understanding of teaching content by doing more examples and exercises.

(D) Teaching content

Section 1 General theory

1, homogeneous linear differential equation

(1) General Solution Construction (Basic Solution Matrix)

(2) judging linear correlation (irrelevance) with lonski determinant.

(3) Two properties of the basic solution matrix.

2. Inhomogeneous equation

(1) General solution structure

(2) Using the constant variation method to find the general solution formula.

Section 2 Linear Differential Equations with Constant Coefficients

1, the introduction and properties of matrix exponential function.

2. The basic solution matrix of homogeneous linear differential equation with constant coefficients is.

3. Using Jordan canonical form to get the basic solution matrix.

4. Find the basic solution matrix by the method of undetermined exponential function.

(1)A has a root.

(2)A has multiple roots

5. Give a special case of the equation that can be solved without Theorems 5 and 6.

The third quarter higher order linear differential equations

1, the general theory of higher order linear differential equations

Construction of general solution of (1) homogeneous linear differential equation.

(2) The introduction of fundamental system of solutions and how to judge it as fundamental system of solutions.

(3) The construction of general solution of nonhomogeneous linear differential equation.

(4) Important examples.

2. Higher order linear differential equations with constant coefficients

(1) The general solution of the homogeneous equation is obtained by using the characteristic roots of the characteristic equation.

(2) For two kinds of special nonhomogeneous equations, how to use their particularity to get special solutions, and then get general solutions.

(3) Examples show that some equations can be transformed into equations for calculation and solution.

(4) Most of the exercises in this section are explained after class.

Third, teaching methods and means

Theories and examples are mainly taught in class. Most of the exercises after class are done by students independently, so it is difficult to explain them during the tutoring time. Supervise students' study, and test their mastery of what they have learned by correcting homework and taking tests in class.

Fourth, curriculum evaluation and requirements

Assessment method: the combination of closed-book written test and usual grades is mastered by the teacher. Take a percentage system.

Verb (abbreviation for verb) the procedure of making a curriculum outline

The outline of this course was finalized by Professor Liang Xiaoli, Dean of Teachers College, Niu Ping, Zhou Yi, Nie Xijun, Li Yanhong and Yu Qiang, Vice Presidents, and Nie Xijun wrote it.

Textbooks and teaching reference materials used in intransitive verbs course

(A) the name of the textbook: ordinary differential equations

(2) References:

1, ordinary differential equation Northeast Normal University

2. Ordinary differential equation of Sun Yat-sen University

3. Ordinary differential equation Ding Chongwen

(3) Other references:

1. Solutions and skills of typical problems of ordinary differential equations Ding Chongwen

2. Exercises and solutions of ordinary differential equations Ding Chongwen

3. Solving ordinary differential equation problems