## 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
Timetable
Wed 14:00–15:50 M3,01023
• Timetable of Seminar Groups:
M7160/01: Wed 16:00–16:50 M3,01023, M. Veselý
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
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
recommended 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
not specified
• 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. 418 s. info
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