Ordinary differential equation

Description: Differential equations of the first order. Differential equations of higher orders allowing a decrease in order. Structure of solving linear homogeneous and inhomogeneous equations. Method variation arbitrary constant. Systems of linear differential equations and basic methods for its solution. The boundary value problem for the linear equation of the second order. The main concepts of the theory of stability. Equations with private first-order derivatives.

Amount of credits: 5

Пререквизиты:

• Mathematical Analysis 2

Course Workload:

Component: University component

Cycle: Base disciplines

Goal

• Formation of students' scientific and practical understanding of mathematical methods for describing and solving practical problems in engineering, technology, and economics.

Objective

• basic methods for solving applied problems in this discipline related to the specialty, actions with various quantities and evaluation of their order;
• approximate methods for solving differential and integral equations, as well as their systems;
• approximate methods of problem analysis and control of the correctness of solutions.

Learning outcome: knowledge and understanding

• Knowledge and understanding of the basic mathematical definitions, theorems and other theoretical information of the course "Differential Equations", as well as knowledge of the types of problems solved by certain mathematical methods

Learning outcome: applying knowledge and understanding

• Application of knowledge and skills in the formulation of applied practical problems by mathematical methods, as well as the use of known methods for solving the formulated problems;

Learning outcome: formation of judgments

• the ability, based on the existing knowledge of the discipline "Differential Equations", to draw conclusions about possible methods for analyzing and solving practical problems in a special area;

Learning outcome: communicative abilities

• ability to work in a team to effectively solve the set practical problems based on knowledge of mathematical methods;

Learning outcome: learning skills or learning abilities

• the ability of independent or on the basis of educational educational programs for advanced training in the field of mathematical knowledge in order to meet the modern requirements of the specialty.

Teaching methods

The main forms of teaching the discipline are thematic lectures, practical classes, independent work of the student under the guidance of a teacher, consultations. The main methods of lecturing are problematic, dialogical, personalized presentations. In visualization lectures, a visual form of presenting lecture material by means of TCO, audio-video equipment, natural objects, models, symbolic visualization, multimedia can be used and is reduced to a detailed or brief commentary by the lecturer on these materials. Practical classes are a group form of training and are aimed at consolidating theoretical material. They solve typical problems and perform exercises on the topics of the course. Practical classes can also be conducted using multimedia and computer equipment and software.

Assessment of the student's knowledge

Teacher oversees various tasks related to ongoing assessment and determines students' current performance twice during each academic period. Ratings 1 and 2 are formulated based on the outcomes of this ongoing assessment. The student's learning achievements are assessed using a 100-point scale, and the final grades P1 and P2 are calculated as the average of their ongoing performance evaluations. The teacher evaluates the student's work throughout the academic period in alignment with the assignment submission schedule for the discipline. The assessment system may incorporate a mix of written and oral, group and individual formats.

The evaluating policy of learning outcomes by work type

Evaluation form

The student's final grade in the course is calculated on a 100 point grading scale, it includes:

• 40% of the examination result;
• 60% of current control result.

The final grade is calculated by the formula:

 

Where Midterm 1, Midterm 2are digital equivalents of the grades of Midterm 1 and 2;

E is a digital equivalent of the exam grade.

Final alphabetical grade and its equivalent in points:

The letter grading system for students' academic achievements, corresponding to the numerical equivalent on a four-point scale:

Topics of lectures

• Уравнения с разделенными и разделяющимися переменными
• Линейные дифференциальные уравнения 1-го порядка
• Дифференциальные уравнения в полных дифференциалах
• Уравнения высших порядков, допускающие понижение порядка
• Линейные однородные дифференциальные уравнения с постоянными коэффициентами
• Линейные неоднородные дифференциальные уравнения с постоянными коэффициентами
• Краевая задача для линейного уравнения
• Системы линейных дифференциальных уравнений и методы их решения
• Основные понятия теории устойчивости
• Численные методы решения задачи Коши для обыкновенных дифференциальных уравнений

Key reading

• Г. Мутанов, Н.Хисамиев, С.Тыныбекова. Проблемно-ориентированный курс дифференциальных уравнений для студентов технических вузов.-Усть-Каменогорск, 2018.
• В.В.Амельсин. Дифференциальные уравнения в приложениях. - М.: Наука, 2013.
• А.Б.Васильева, А.Н.Тихонов. Интегральные уравнения.– М.: ФИЗМАТЛИТ, 2018.
• А.Н.Тихонов. Дифференциальные уравнения: учебник для вузов.- М.:Лань, 2018.
• Берман Г.Н. Сборник задач по курсу математического анализа. – М.: ФИЗМАТЛИТ, 2019.
• Демидович Б.П. Сборник задач и упражнений по математическому анализу: учебное пособие.-М.:Астрель-АСТ,2019.
• А.Б.Васильева, Г.Н.Медведев, А.Н.Тихонов. Дифференциальные и интегральные уравнения, вариационное исчисление в примерах и задачах. –М.:ФИЗМАТЛИТ,2013.
• Тыныбекова С.Д. Дифференциальные и интегральные уравнения. - Усть-Каменогорск, 2013.

Further reading

• Данко И.Е., Попов А.Г., Кожевникова Т.Я. Высшая математика в упражнениях и задачах. – М.: Высшая школа, 2018 ч.1,2.
• Кузнецов Л.А. Сборник задач по высшей математике (типовые расчеты). – М.: Высш. школа, 2017.
• Степанов, В. В. Курс дифференциальных уравнений / В. В. Степанов. - М.:Лань, 2018. - 468 с.
• Кудрявцев Л.Д. Математический анализ. Учебник для вузов. М., Высш. шк., 2017.
• Филиппов, А. Ф. Сборник задач по дифференциальным уравнениям : Учеб. пособие для вузов / А.Ф. Филиппов. - 3-е изд. стереотип. - М.:Лань, 2018. - 96 с.
• Понтрягин, Л. С. Обыкновенные дифференциальные уравнения [Текст] : учеб. для гос. ун-тов / Л.С. Понтрягин. - 3-е изд., стереотип. - М.:Лань, 2018. - 332 с.