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Бакалавриат 2022/2023

# Дифференциальные уравнения

Статус: Курс обязательный (Прикладной анализ данных)
Направление: 01.03.02. Прикладная математика и информатика
Когда читается: 2-й курс, 3, 4 модуль
Формат изучения: без онлайн-курса
Охват аудитории: для своего кампуса
Преподаватели: Букин Кирилл Александрович, Гончаренко Василий Михайлович, Киреенков Алексей Альбертович, Протасов Владимир Юрьевич
Язык: английский
Кредиты: 4
Контактные часы: 72

### Course Syllabus

#### Abstract

“Differential Equations” is a spring semester course for second-year students studying at the Faculty of Computer Sciences. It is designed to suit the requirements of the Faculty of Computer Sciences curriculum as well as UoL where DE is a part of the mathematical curriculum. Besides the course on differential equations is included as a topic in “Mathematical methods for economists” external exam. This course is an important part of the bachelor stage in education of the future applied mathematicians and computer scientists. It has to give students skills for implementation of mathematical knowledge and expertise. Its prerequisite is the knowledge of the single variable calculus.

#### Learning Objectives

• The knowledge given during this course will allow the students to progress with further complicated topics such as optimal control theory and like.
• This course will give the students skills for the implementation of mathematical knowledge and expertise.

#### Expected Learning Outcomes

• Students will be able to solve linear equations with the constant coefficients as well as the systems of such equations.
• Students will develop an understanding of basic concepts of the differential and difference equations.
• Students will develop the ability to apply the knowledge of the differential and difference equations which will enable them to analyze dynamics of the processes.
• Students will develop the ability to apply the knowledge of the differential and difference equations which will enable them to analyze dynamics of the processes.

#### Course Contents

• Dynamics in economics and natural sciences. Simple first-order equations. Separable equations. Concept of stability of the solution of ODE. Exact equations. General solution as a sum of a general solution of homogeneous equation and a particular solution of a nonhomogeneous equation. Bernoulli equation. Fundamental theorem on existence and uniqueness.
• Qualitative theory of differential equations. Solow’s growth model from macroeconomics.
• Second-order linear differential equations with constant coefficients.
• Refresher on complex numbers and operations on them. Representation of a number. De Moivre and Euler formulas.
• Higher-order linear differential equation with constant coefficients. Characteristic equation. Method of undetermined coefficients for the search of a particular solution. Stability of solutions. Routh theorem (without proof). Systems of DE (linear equations case). Variation of parameters method. Solving linear equations with the variable coefficients.
• Discrete time economic systems. Difference equations. Method of solving first-order equations. Convergence and oscillations of a solution. Cobweb model. Partial equilibrium model with the inventory.
• Second-order difference equations.
• Higher-order difference equations. Characteristic equation. Undetermined coefficients method. Conditions for the stability of solutions. Markov processes.
• Stability of linear systems via eigenvalues. Stability of nonlinear systems.
• Phase portraits of planar systems.
• First integrals.
• Liapunov functions.

#### Assessment Elements

• Homework
• In-class assignment
• Exam

#### Interim Assessment

• 2022/2023 4th module
0.3 * In-class assignment + 0.3 * Homework + 0.4 * Exam

#### Recommended Core Bibliography

• Mathematics for economists, Simon, C. P., 1994