Differential Equations

2024/2025

Учебный год

ENG

Обучение ведется на английском языке

4

Кредиты

Автор программы

Протасов Владимир Юрьевич

Кто читает:

Департамент больших данных и информационного поиска

Статус:

Курс обязательный

Когда читается:

2-й курс, 3, 4 модуль

Преподаватели

Башминова Дарья Александровна

Буитраго Оропеса Диего Фернандо

Буитраго Оропеса Хуан Карлос

Гончаренко Василий Михайлович

Жуков Павел Владимирович

Мажуга Андрей Михайлович

Тараканов Александр Александрович

Томащук Корней Кириллович

Course Syllabus

Abstract

This course provides a rigorous and application-oriented introduction to ordinary differential equations as a mathematical framework for modeling, analysis, and interpretation of dynamical processes in data science, business analytics, and machine learning. The course covers theoretical foundations such as existence and uniqueness of solutions, dependence on initial conditions and parameters, classical solution methods, linear systems and matrix exponentials, Laplace transform, variational principles, and stability analysis of dynamical systems.

Learning Objectives

• The main purpose of this course is to provide students with a solid theoretical and applied understanding of ordinary differential equations as a tool for modeling and analyzing dynamical systems relevant to data science and business analytics.

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.
• 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.
• Qualitative theory of differential equations. Dynamical Systems.

Assessment Elements

• Homework
• Midterm exam
• Exam

Interim Assessment

• 2024/2025 4th module
  0.4 * Exam + 0.3 * Homework + 0.3 * Midterm exam

Bibliography

Recommended Core Bibliography

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

Recommended Additional Bibliography

• Курс дифференциальных уравнений и вариационного исчисления : учеб. пособие для вузов, Романко, В. К., 2001
• Сборник задач по дифференциальным уравнениям и вариационному исчислению, Романко, В. К., 2002

Authors

• Tarakanov Aleksandr Aleksandrovich
• Bukin Kirill Aleksandrovich