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Course, academic year 2023/2024
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Ordinary Differential Equations - NMMA336
Title: Obyčejné diferenciální rovnice
Guaranteed by: Department of Mathematical Analysis (32-KMA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2022
Semester: summer
E-Credits: 5
Hours per week, examination: summer s.:2/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech
Teaching methods: full-time
Teaching methods: full-time
Additional information:
Guarantor: Oleksandr Minakov, Ph.D.
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, NMMA333
Interchangeability : NDIR012, NDIR020, NMMA333
Is incompatible with: NMMA333
Is pre-requisite for: NMNM351, NMMA351
Is interchangeable with: NMMA333
Annotation -
Last update: doc. Mgr. Petr Kaplický, Ph.D. (29.05.2019)
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. Mgr. Petr Kaplický, Ph.D. (29.05.2019)

Basic lectures on ordinary differential equations. Basic knowledge of linear algebra, mathematical analysis and integration theory is required.

Course completion requirements -
Last update: doc. Mgr. Petr Kaplický, Ph.D. (29.05.2019)

The credit for exercises is awarded either for activity at the exercises or for solving a given set of problems (each student can choose).

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. Mgr. Petr Kaplický, Ph.D. (29.05.2019)

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. Mgr. Petr Kaplický, Ph.D. (29.05.2019)

Ability to solve problem similar to those solved at the exercises, knowledge of the theory presented in the lecture, understanding. Details at (in czech).

Syllabus -
Last update: doc. Mgr. Petr Kaplický, Ph.D. (29.05.2019)


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.


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.


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


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


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


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


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

Entry requirements -
Last update: doc. Mgr. Petr Kaplický, Ph.D. (29.05.2019)

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

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