M7160 Ordinary Differential Equations II

Faculty of Science
Spring 2011
Extent and Intensity
2/1/0. 3 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
Teacher(s)
prof. Alexander Lomtatidze, DrSc. (lecturer)
Guaranteed by
doc. RNDr. Josef Kalas, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Timetable
Tue 12:00–13:50 MS1,01016
  • Timetable of Seminar Groups:
M7160/01: Tue 14:00–14:50 MS1,01016, A. Lomtatidze
Prerequisites
M5160 Ord. Differential Equations I
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. Differential equations: Linear and nonlinear systems of ordinary differential equations, existence, uniqueness and properties of solutions, elements of the 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
  • Mathematics (programme PřF, M-MA, specialization Mathematical Analysis)
  • Mathematics (programme PřF, N-MA, specialization Mathematical Analysis)
Course objectives
The theory of differential equations ranks among basic parts of mathematical analysis. The course is focused to systems of nonlinear differential equations with a Carathéodory right-hand side. The following questions are studied in detail: the existence of a solution of the Cauchy problem, extendibility of solutions, global solutions, structure of a solution set of the Cauchy problem, continuous dependence of solutions on parameters. At the end of the course students should be able to: 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; to analyse selected problems from the topics of the course.
Syllabus
  • Carathéodory class of functions
  • On absolutly continuous functions
  • Cauchy problem
  • Carathéodory theorem for higher-order differential equations
  • Extendibility of solutions of the Cauchy problem
  • Lower and upper solutions of the Cauchy problem
  • On a set of solutions of the Cauchy problem
  • Existence of lower and upper solutions
  • Theorems on differential inequalities
  • Theorems on integral inequalities
  • Global solvability of the Cauchy problem
  • Uniqueness of a solution
  • Correctness of the Cauchy problem
  • Structure of a set 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
  • KALAS, Josef and Miloš RÁB. Obyčejné diferenciální rovnice. 2. vyd. Brno: Masarykova univerzita, 2001, 207 s. ISBN 8021025891. 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
  • 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
  • CODDINGTON, Earl A. and Norman LEVINSON. Theory of ordinary differential equations. New York: McGraw-Hill, 1955, 429 s. info
Teaching methods
lectures and class exercises
Assessment methods
Teaching: lecture 2 hours a week, seminar 1 hour a week. Examination: written and oral.
Language of instruction
Czech
Further comments (probably available only in Czech)
Study Materials
The course is taught once in two years.
The course is also listed under the following terms Spring 2011 - only for the accreditation, Autumn 2000, Autumn 2002, Autumn 2004, Autumn 2006, Autumn 2008, Spring 2014, Spring 2016, spring 2018, Spring 2020, Spring 2022, Spring 2024.
  • Enrolment Statistics (Spring 2011, recent)
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