Differential Equations

2026/2027

Academic Year

ENG

Instruction in English

4

ECTS credits

Delivered at:

Big Data and Information Retrieval School

Course type:

Compulsory course

When:

2 year, 3, 4 module

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
• First-Order ODEs: Basic Exactly Solvable Equations
• First-Order ODEs: Exact Equations and Transformations
• Higher-Order ODEs and Reduction Techniques
• Phase Portraits and Qualitative Analysis of Nonlinear ODEs
• Linear ODEs
• Systems of Linear ODEs and Difference Equations
• Matrix Exponent
• Method of Variation of Constants
• Linear Finite Difference Equations and Discrete Dynamical Systems
• Bifurcations and Nonlinear Dynamical Behavior
• Laplace Transform
• Stability of Linear Systems
• Lyapunov Functions and Stability
• Linearization and Stability
• Geometry of State Spaces and Calculus on Surfaces
• Variational Problems and Functionals
• Principle of Least Action and Euler–Lagrange Equations
• Numerical Methods for Solving ODEs

Assessment Elements

• quiz
• final exam
• colloquium
• homework

Interim Assessment

• 2026/2027 4th module
  Final = max(0.25 * quiz + 0.25 * colloquium + 0.5 * final_exam, 0.5 * homework)

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