Ordinary Differential Equations - NMMA333
Title: Obyčejné diferenciální rovnice
Guaranteed by: Department of Mathematical Analysis (32-KMA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2021
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: not taught
Language: Czech
Teaching methods: full-time
Teaching methods: full-time
Additional information: https://dl1.cuni.cz/course/view.php?id=10150
Class: M Bc. OM
M Bc. OM > Zaměření MA
M Bc. OM > Zaměření NUMMOD
M Bc. OM > Povinně volitelné
Classification: Mathematics > Differential Equations, Potential Theory
Incompatibility : NDIR012, NDIR020, NMMA336
Interchangeability : NDIR012, NDIR020, NMMA336
Is incompatible with: NMMA336, NDIR012, NDIR020
Is pre-requisite for: NMMA349, NMNM349
Is interchangeable with: NDIR020, NMMA336, NDIR012
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download Anotace ODR 1.doc doc. RNDr. Jaroslav Milota, CSc.
Annotation -
Last update: G_M (16.05.2012)
A course for bachelor's program in General Mathematics. Recommended for specializations Mathematical Analysis and Mathematical Modelling and Numerical Analysis.
Aim of the course -
Last update: doc. RNDr. Jaroslav Milota, CSc. (29.08.2012)

Basic lectures on ordinary differential equations. Basic knowledge of linear algebra, mathematical analysis and integration theory is required.
Course completion requirements -
Last update: doc. RNDr. Tomáš Bárta, Ph.D. (29.09.2020)

The credit for exercises is awarded for activity at the exercises and for solving homeworks.

The credit from exercises is required to participate at the exam.

The exam consists of a written computational part (test) and oral theoretical part.

Students failing in the test are not allowed to continue with the oral part. Students failing in the oral part must go through both parts (written and oral) in their next attempt.
Literature -
Last update: doc. RNDr. Tomáš Bárta, Ph.D. (23.05.2019)

M. Braun: Differential equations and their applications. QA371.B795 1993

I.I. Vrabie: Differential equations: an introduction to basic concepts, results, and applications. QA371.V73 2004

Requirements to the exam -
Last update: doc. RNDr. Tomáš Bárta, Ph.D. (05.11.2020)

Ability to solve problem similar to those solved at the exercises, knowledge of the theory presented in the lecture, understanding. Study materials available in moodle (write to the lecturer to get the password).
Syllabus -
Last update: doc. RNDr. Tomáš Bárta, Ph.D. (21.04.2015)

1.

Peano Theorem on local existence of solutions, local and global uniqueness, sufficient conditions for local uniqueness. Maximal solution - existence, characterisation. Gronwall lemma. Continuous and differentiable dependence of solutions on parameters or initial value.

2.

Linear equations: global existence and uniquness. Fundamental matrix, Wronskian, Liouville's formula. Variation of parameters in integral form. Linear systems with constant coefficients. Exponential of a matrix and its properties. Stable, unstable and central subspaces.

3.

Stability, asymptotic stability. Uniform stability. Stability of linear equations. Linearized stability and unstability.

4.

First integral, orbital derivative. Existence of first integrals. Application: method of characteristics.

5.

Higher order equations: reformulation as a first order system. Theorems on local existence and uniquness. Variation of parameters.

6.

Stability II: Lyapunov function, theorems on stability and asymptotic stability. Ljapunov equation.

7.

Floquet theory: logaritm of a matrix. Existence of periodic solutions and their stability.

Entry requirements -
Last update: doc. RNDr. Tomáš Bárta, Ph.D. (23.05.2019)

Differential and integral calculus, linear algebra (matrix calculus, eigenvalues and eigenvectors, etc.)