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PřF:M7160 Ord. Differential Equations II - Course Information

## M7160 Ordinary Differential Equations II

**Faculty of Science**

Spring 2022

**Extent and Intensity**- 2/1/0. 5 credit(s). Type of Completion: zk (examination).

Taught in person. **Teacher(s)**- doc. RNDr. Michal Veselý, Ph.D. (lecturer)
**Guaranteed by**- doc. RNDr. Michal Veselý, Ph.D.

Department of Mathematics and Statistics - Departments - Faculty of Science

Supplier department: Department of Mathematics and Statistics - Departments - Faculty of Science **Prerequisites**-
**M5160**Ord. Differential Equations I

Mathematical analysis: Differential calculus of functions of several variables, integral calculus, sequences and series of numbers and functions, metric spaces, complex functions of a real variable.

Linear algebra: Systems of linear equations, determinants, linear spaces, linear transformations and matrices, canonical form of a matrix.

Differential equations: Linear and non-linear systems of ordinary differential equations, stability theory, autonomous equations. **Course Enrolment Limitations**- The course is also offered to the students of the fields other than those the course is directly associated with.
**fields of study / plans the course is directly associated with**- Mathematical Analysis (programme PřF, N-MA)

**Course objectives**- The course is focused on systems of non-linear differential equations with the Carathéodory right-hand side. The following questions are studied in detail: the existence of solutions of the Cauchy problem; the extendibility of solutions; and the existence of global solutions.
**Learning outcomes**- At the end of the course, students will be able to:

define and interpret the basic notions used in the mentioned fields;

formulate relevant mathematical theorems and statements and to explain methods of their proofs;

use effective techniques utilized in the subject areas;

analyse selected problems from the topics of the course. **Syllabus**- The Carathéodory class of functions
- Absolutely continuous functions
- The Carathéodory theorem for higher-order differential equations
- Extendibility of solutions of the Cauchy problem
- Lower and upper solutions of the Cauchy problem
- Set of solutions of the Cauchy problem
- Differential and integral inequalities
- Global solutions of the Cauchy problem
- Uniqueness of solutions of the Cauchy problem

**Literature**- HARTMAN, Philip.
*Ordinary differential equations*. 2nd ed. Philadelphia, Pa.: SIAM, 2002. xx, 612 s. ISBN 0-89871-510-5. info - CODDINGTON, Earl A. and Norman LEVINSON.
*Theory of ordinary differential equations*. New York: McGraw-Hill, 1955. 429 s. info - KIGURADZE, Ivan.
*Okrajové úlohy pro systémy lineárních obyčejných diferenciálních rovnic*. 1. vyd. Brno: Masarykova univerzita, 1997. 183 s. ISBN 80-210-1664-7. info

*recommended literature*- KALAS, Josef and Miloš RÁB.
*Obyčejné diferenciální rovnice*. 2. vyd. Brno: Masarykova univerzita, 2001. 207 s. ISBN 8021025891. info - KURZWEIL, Jaroslav.
*Obyčejné diferenciální rovnice : úvod do teorie obyčejných diferenciálních rovnic v reálném oboru*. 1. vyd. Praha: SNTL - Nakladatelství technické literatury, 1978. 424 s. info

*not specified*- HARTMAN, Philip.
**Teaching methods**- Lectures, seminars
**Assessment methods**- The final oral exam (60 minutes) for 20 points. For successfull examination (the grade at least E), the student needs 10 points or more.
**Language of instruction**- Czech
**Further Comments**- The course is taught once in two years.

The course is taught: every week.

- Enrolment Statistics (Spring 2022, recent)
- Permalink: https://is.muni.cz/course/sci/spring2022/M7160