Bachelor 2020/2021

Differential Equations

Type: Compulsory course (HSE University and University of London Double Degree Programme in Data Science and Business Analytics)

Area of studies: Applied Mathematics and Information Science

Delivered by: Big Data and Information Retrieval School

Where: Faculty of Computer Science

When: 2 year, 3, 4 module

Mode of studies: offline

Instructors: Kirill Bukin, Vasily Goncharenko, Vladimir Protasov

Language: English

ECTS credits: 5

Contact hours: 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. In the spring semester the course is split into two unequal parts: it is taught from January through the end of April and after the students finish with their UoL exams in May it will resume and will continue till mid-June. The assessment of the students will be done by setting mock exam by the end of the 3rd module, then later by the University of London (UoL) examinations in May and final exam will be set in late June. But the final grade will depend solely on mock, final exams performance and home assignments grades.

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 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 an understanding of basic concepts of the differential and difference equations.
• Students will be able to solve linear equations with the constant coefficients as well as the systems of such equations.

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

• Home assignments
  Every second week
• Midterm Test
• Final Exam
  The exam may be carried out online via distance learning platforms. Письменный экзамен продолжительностью в 120 минут с асинхроннмы прокторингом. Экзамен выполняется на бумаге, решения студентами фотографируются или сканируются и загружаются в Экзамусе. 30 июня возможно проведение дополнительного мероприятия для тех, кто по техническим причинам не смог пройти экзамен в основную дату. На дополнительном контроле задания будут отличаться.

Interim Assessment

• Interim assessment (4 module)
  0.4 * Final Exam + 0.3 * Home assignments + 0.3 * Midterm Test

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