Mathématique

I. The content of science

        Objective of teaching the subject: The main content of the differential equations course is to introduce students to the concepts of first -order ordinary differential equations , higher order differential equations, system of 1st order linear differential equations , To develop skills in creating mathematical support using methods for approximate solution of differential equations and Laplace transforms .

serves to teach students to think logically, draw correct conclusions, and increase their mathematical culture, while at the same time introducing students to the foundations of modern mathematics, consciously researching issues related to professional activity, and explaining the essence of the possibilities of calculus in solving problems and teaching them to apply them .

 The tasks of teaching science are: The science of differential equations is about acquiring a culture of thinking through the conscious mastery of the content, rules, and methods of mathematical concepts, understanding, generalizing, and analyzing information, setting goals, and choosing ways to achieve them;

- express their thoughts logically, clearly and rationally, justifying their oral and written speech;

-to achieve competencies in applying the basic methods of the science of differential equations, including theoretical and experimental research methods, to professional activities, and thereby to understand the importance of linear algebra in the study of programming, economics and other fields, in solving their problems, especially in the analysis of mathematical models of various processes.

II. Main theoretical part (lectures)

Science content following topics includes :

Topic 1. Differential equation. Basic concepts and definitions.

Concepts about differential equations. Problems leading to differential equations . Differential equations with a separable variable .

Topic 2. Variables detachable and to him/her coming differential equations .

Variables​ detachable differential equations . Variables detachable differential to the equation coming equations.

Topic 3. Homogeneous and homogeneous differential equations. Application to practical problems (Reflection problem) .

Same -sex differential equations . Homogeneous differential to the equation coming equations . Application to practical problems (Reflection problem) .

Topic 4. Linear differential equations and their properties .

Linear differential equation .​ solution , integral multiplier using solution .​ to solve , to undo variation method .

Topic 5. Linear differential to the equation coming equations . Bernoulli and Ricatti equation

Equations leading to linear differential equations . Bernoulli equation . Ricciardi equation .

Topic 6. Complete differential equation .

Complete differential equation . Complete differential equation solution​

Topic 7. First order differential equation for the Cauchy problem . Solution existence and uniqueness

First order differential equation Cauchy problem for . Cauchy problem solution existence . Cauchy problem solution uniqueness .

Topic 8. Differential equations not solved with respect to the derivative . Lagrange and Clairault equations .

To the harvest relatively undefeated equation . with respect to y reveal differential equation . with respect to x reveal differential equation . Claret and Lagrange equation .

Topic 9. Higher-order differential with decreasing order equations .

High orderly differential equations . Order consecutively decreasing equations . Order k to the bottom decreasing equations . Order for one decreasing equations .

Topic 10. Higher-order linear differential equations. Basic theorems .

Solution funny freedom . Fundamental system of solutions . Quadrature area . Surface additivity .

Topic 11. Linear homogeneous with constant coefficients Differential equations. Characteristic polynomial .

Unchangeable with coefficient linear differential equations . Homogeneous linear differential equation solution . Homogeneous not linear differential equation solution​

Topic 12. Nonhomogeneous differential equations with constant coefficients, the right-hand side of which is in a special form .

Nonhomogeneous differential equations with constant coefficients, the right-hand side of which is in a special form .

Topic 13. Higher -order linear differential equations with constant coefficients. Higher - order linear differential equations with constant coefficients.

Topic 14. Second-order differential equations and them The method of solving the invariant by variation. Ostrogradsky-Liouville formula .

Second-order differential equations and them The method of solving the invariant by variation. Ostrogradsky-Liouville formula .

Topic 15. Methods for approximate solution of differential equations (using mathematical packages) .

Methods for approximate solution of differential equations using mathematical packages .

Topic 16. Systems of differential equations. Solution methods .

Linear differential equations system . Homogeneous linear differential equations system solution . Homogeneous not linear differential equations system solution​

Topic 17. Original and image. Laplace transform .

Original and image. Laplace transform .

Topic 18. Basic properties of the Laplace transform .

Basic properties of the Laplace transform .

Topic 19. . A system of linear equations in matrix form.

A system of linear equations in matrix form.

Topic 20 . A system of non-homogeneous equations with a C h line.

C h line.

Topic 21. . Operational calculus method for solving systems of differential equations .

Operational calculus method for solving systems of differential equations .

Instructions and recommendations for practical trining

     Practical training should be conducted by one professor per academic group. It is advisable that the training be conducted using active and interactive methods, and appropriate pedagogical and information technologies should be used.

The following topics are recommended for practical training:

Topic 1. Variables detachable and to him/her coming differential equations

Topic 2 . Homogeneous and homogeneous differential equations.

Topic 3. Linear differential equations and their properties .

Topic 4. Linear differential to the equation coming equations . Bernoulli and Ricatti equation

Topic 5. Full differential equation .

Topic 6. First order differential equation for the Cauchy problem . Solution existence and uniqueness

Topic 7. Differential equations that are not solved with respect to the derivative . Lagrange and Clairault equations .

Topic 8. Higher-order differential with decreasing order equations .

Topic 9. Higher order linear differential equations.

Topic 10.  Linear homogeneous with constant coefficients Differential equations. Characteristic polynomial .

Topic 11. Nonhomogeneous differential equations with constant coefficients, the right-hand side of which is in a special form .

Topic 12. Second-order differential equations and them The method of variationalization of the invariant. The Ostrogradsky-Liouville formula . Methods for approximate solution of differential equations (using mathematical packages) .

Topic 13. Systems of differential equations. Methods of solution .

Topic 14. Original and image. Laplace transform . Basic properties of the Laplace transform .

Topic 15. Operational calculus method for solving differential equations and systems of differential equations .

IV . Independent learning and independent work

The content of independent learning is determined by students

- preparation for lectures and practical exercises;

- doing homework;

- mastering theoretical knowledge;

- performing differentiated individual tasks;

- consists of mastering topics intended for independent study.

Recommended topics for independent study:

1. To the harvest relatively untied equations .
2. Special points and special solutions .
3. Full differential equation , integrator multiplier​
4. The Koshi problem solution existence and uniqueness .
5. Procedure decreasing differential equations .
6. His solution , general , private solutions .
7. Unchangeable with coefficient linear , one gender equations .
8. Special right sided Linear equations with n-order constant coefficients solution​
9. Information about systems of differential equations.
10. Differential equation geometry and to physics applications .

V. Science education results ( formed) competencies )

By science students knowledge , skill and qualifications following requirements Student :

– First-order ordinary differential equations , higher order differential equations, system of 1st order linear differential equations , About methods for approximate solutions of differential equations, methods of working using Laplace transforms to knowledge has to be ;

– First -order ordinary differential equations , higher order differential equations, system of 1st order linear differential equations , methods for approximate solutions of differential equations, using Laplace transforms methods​ application skills has to be ;

– first-order ordinary differential equations , higher order differential equations, system of 1st order linear differential equations , methods for approximate solutions of differential equations, using Laplace transforms examples solution ;

VI. Educational technologies and methods :

• lectures;
• interactive case studies;
• seminars (logical reasoning, quick questions and answers);
• working in groups;
• making presentations;
• individual projects;
• projects to work and defend as a team .

VII. Requirement for obtaining loans:

Fully master the theoretical and methodological concepts of the subject, be able to correctly reflect the results of the analysis, conduct independent observations of the processes being studied, and complete the tasks and assignments given in the forms of current and intermediate control, and submit a written work for final control.

Main literature

1.  Stepanov V.V. Kurs differentialnyx uravneniy, M., KomKniga. USSR. 2006.
2. Robinson JC An Introduction to Ordinary Differential Equations, Cambridge University Press 2013 .
3. Filippov A. F. Sbornik zadach po differentialnym uravneniyam. Izhevsk: Iz-vo RXD. 2000 .
4. Khurramov Sh. R. « Higher Mathematics . Volumes 1-2 . Tashkent, "Tafakkur" Publishing House, 201 8.

Additional literature

1. Jabborov NM « Higher Mathematics . Parts 1-2 . Karshi , 2010 .
2. Rakhmatov R., Tadzhibayeva Sh.E., Shoyimardonov SK Higher Mathematics. Volume 1. 2017.
3. Piskunov N.S. Differential and integral work ­len ie for VTUZov. - M.: Nauka, v 2x chastyakh, 2001.

Internet resources

1.     www.tdpu.uz 

2 .    www.pedagog.uz

3 .    www.edu.uz

4.     www.nadlib.uz (Uzbek Library named after A. Navoi)