Differential Equations

2025/2026

Учебный год

ENG

Обучение ведется на английском языке

4

Кредиты

Кто читает:

Департамент больших данных и информационного поиска

Статус:

Курс обязательный

Когда читается:

2-й курс, 3, 4 модуль

Преподаватели

Башминова Дарья Александровна

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

• Formulate mathematical models of dynamical processes using ordinary differential equations.
• Analyze existence and uniqueness of solutions to initial value problems.
• Evaluate sensitivity of solutions with respect to initial conditions and model parameters.
• Apply classical analytical methods to solve basic classes of ordinary differential equations.
• Derive equations of motion using the principle of least action and Euler–Lagrange equations
• Analyze stability of equilibria using linearization and Lyapunov methods.

Course Contents

• Introduction to Ordinary Differential Equations, Existence and Uniqueness of Solutions
• Continuity and Differentiability with Respect to Parameters
• Classical Examples of Exactly Solvable ODE 1
• Classical Examples of Exactly Solvable ODE 2
• Classical Examples of Exactly Solvable ODE 3
• Basics of Complex Numbers
• Linear ODEs
• Systems of Linear ODEs and Difference Equations
• Matrix Exponent
• Method of Variation of Constants
• Linear Finite Difference Equations
• System of Linear Finite Difference Equations
• Laplace Transform
• Stability of Linear Systems
• Lyapunov Functions and Stability
• Linearization and Stability
• Dynamical Systems and Equilibrium Points
• Variational Problems and Functionals
• Principle of Least Action and Euler–Lagrange Equations
• Numerical Methods for Solving ODEs

Assessment Elements

• quiz
• homework
• colloquium
• final_exam

Interim Assessment

• 2025/2026 4th module
  0.25 * colloquium + 0.35 * final_exam + 0.2 * homework + 0.2 * quiz

Bibliography

Recommended Core Bibliography

• Mathematics for economists, Simon, C. P., 1994
• Ordinary differential equations, Birkhoff, G., 2004

Recommended Additional Bibliography

• Курс дифференциальных уравнений и вариационного исчисления : учеб. пособие для вузов, Романко, В. К., 2011
• Сборник задач по дифференциальным уравнениям и вариационному исчислению, Романко, В. К., 2015

Authors

• Tarakanov Aleksandr Aleksandrovich