M5160 Ordinary Differential Equations I

Faculty of Science
Autumn 2018
Extent and Intensity
2/2/0. 4 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
Teacher(s)
doc. RNDr. Josef Kalas, CSc. (lecturer)
Guaranteed by
doc. RNDr. Josef Kalas, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Supplier department: Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Mon 17. 9. to Fri 14. 12. Wed 13:00–14:50 M4,01024
  • Timetable of Seminar Groups:
M5160/01: Mon 17. 9. to Fri 14. 12. Wed 15:00–16:50 M4,01024, J. Kalas
Prerequisites
M3100 Mathematical Analysis III && M2110 Linear Algebra II
Mathematical analysis: Differential calculus of functions of one and several variables, integral calculus, sequences and series of numbers and functions, metric spaces, complex function of a real variable. Linear algebra: Systems of linear equations, determinants, linear spaces, linear transformation and matrices, canonical form of a matrix.
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 theory of differential equations ranks among fundamental parts of mathematical analysis. It is utilized by a number of other courses and in many applications. The basic aim of the course is to familiarize students with the fundamentals of the theory of ordinary differential equations, with the basic parts of the stability and qualitative theory of differential equations and to show connections with other parts of mathematics.
After passing the course, the student will be able:
to define and interpret the basic notions used in the fields mentioned above;
to formulate relevant mathematical theorems and statements and to explain methods of their proofs;
to use effective techniques utilized in these subject areas;
to apply acquired pieces of knowledge for the solution of specific problems.
Syllabus
  • 1. Fundamental notions - ordinary differential equations and their systems, order of an equation, initial value problem, solutions of differential equations and initial value problems. 2. Systems of linear differential equations - existence and uniqueness of solutions, structure of the family of solutions, variation-of-constants method, linear systems with constant coefficients, connection of linear systems with higher-order linear differential equations. 3. Local and global properties of solutions - local existence and uniqueness of solutions of nonlinear initial value problems, global existence and uniqueness, dependence of solutions on initial values and parameters. 4. Introduction to the stability theory - Lyapunov concept of stability, uniform, asymptotic and exponential stability, stability of linear and perturbed linear systems, Hurwitz criterion, direct method of Lyapunov.
Literature
    required literature
  • KALAS, Josef and Miloš RÁB. Obyčejné diferenciální rovnice (Ordinary differential equations). Vyd. 3. Brno: Masarykova univerzita, 2012, 207 s. ISBN 9788021058156. info
    recommended literature
  • 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
  • GREGUŠ, Michal, Marko ŠVEC and Valter ŠEDA. Obyčajné diferenciálne rovnice. 1. vyd. Bratislava: Alfa, vydavateľstvo technickej a ekonomickej literatúry, 1985, 374 s. info
  • RÁB, Miloš. Metody řešení diferenciálních rovnic. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1989, 68 s. info
  • RÁB, Miloš. Metody řešení diferenciálních rovnic. Vyd. 1. Praha: Státní pedagogické nakladatelství, 1989, 61 s. info
    not specified
  • Hartman, Philip. Ordinary differential equations. Wiley, New York-London-Sydney, 1964.
  • Coppel, W. A. Stability and asymptotic behaviour of differential equations. D. C. Heath and company, Boston, 1965.
Teaching methods
lectures and class exercises
Assessment methods
One written test will be realized during the semester. It is required to obtain at least half of the total amount of points. Class exercises are obligatory, at most three absences can be excused. The exam is composed of a written and an oral part. The written part consists of four exercises. It is necessary to obtain at least 1.5 from possible 4 points.
Language of instruction
Czech
Follow-Up Courses
Further comments (probably available only in Czech)
Study Materials
The course is taught annually.
The course is also listed under the following terms Autumn 2007 - for the purpose of the accreditation, Autumn 1999, Autumn 2010 - only for the accreditation, Autumn 2001, Autumn 2002, Autumn 2003, Autumn 2004, Autumn 2005, Autumn 2006, Autumn 2007, Autumn 2008, Autumn 2009, Autumn 2010, Autumn 2011, Autumn 2011 - acreditation, Autumn 2012, Autumn 2013, Autumn 2014, Autumn 2015, Autumn 2016, autumn 2017, Autumn 2019, Autumn 2020, autumn 2021, Autumn 2022, Autumn 2023, Autumn 2024.
  • Enrolment Statistics (Autumn 2018, recent)
  • Permalink: https://is.muni.cz/course/sci/autumn2018/M5160