Mathematical Analysis (FSI-UMA-A)

Academic year 2026/2027
Supervisor: doc. Ing. Jiří Šremr, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: English
Aims of the course unit:
 
Learning outcomes and competences:
 
Prerequisites:
 
Course contents:

Attendance at lectures and seminars is obligatory and checked. Absence may be compensated based on an agreement with the teacher.

Course-unit credit is awarded on the following conditions: A semestral project consisting of assigned problems. Active participation in seminars.

Examination: The exam tests the knowledge of definitions and theorems (especially the ability of their application to the given problems) and practical skills in solving particular problems. The exam has written and oral part. For the written exam, one sheet of A4 hand-written paper (two-sided) is permitted with formulas and criteria of your choice (without particular examples). The use of a (simple) calculator is also allowed, but phones and computers are not permitted. The list of topics for the oral part of the exam will be announced at the end of the semestr.

The final grade reflects the result of the examinational test (maximum 90 points), discussion about the semestral project (maximum 10 points).

The grading scheme is as follows: excellent (90-100 points), very good (80-89 points), good (70-79 points), satisfactory (60-69 points), sufficient (50-59 points), failed (0-49 points).
Teaching methods and criteria:
 
Assesment methods and criteria linked to learning outcomes:

Attendance at lectures and seminars is obligatory and checked. Absence may be compensated based on an agreement with the teacher.

Course-unit credit is awarded on the following conditions: A semestral project consisting of assigned problems. Active participation in seminars.

Examination: The exam tests the knowledge of definitions and theorems (especially the ability of their application to the given problems) and practical skills in solving particular problems. The exam has written and oral part. For the written exam, one sheet of A4 hand-written paper (two-sided) is permitted with formulas and criteria of your choice (without particular examples). The use of a (simple) calculator is also allowed, but phones and computers are not permitted. The list of topics for the oral part of the exam will be announced at the end of the semestr.

The final grade reflects the result of the examinational test (maximum 90 points), discussion about the semestral project (maximum 10 points).

The grading scheme is as follows: excellent (90-100 points), very good (80-89 points), good (70-79 points), satisfactory (60-69 points), sufficient (50-59 points), failed (0-49 points).
Controlled participation in lessons:
 
Type of course unit:
    Lecture  13 × 3 hrs. compulsory                  
    Exercise  13 × 2 hrs. compulsory                  
Course curriculum:
    Lecture

Systems of first-order ordinary differential equations (ODE). The existence and uniqueness of a solution to the initial value problem. General solutions of homogeneous and non-homogeneous linear systems.
Methods of solving homogeneous systems of linear ODEs with constant coefficients.
Solving non-homogeneous systems of linear ODEs with constant coefficients - variation of parameters.
Stability of solutions to ordinary differential equations and their systems. Basic notions. Stability of linear systems of ODEs with constant coefficients.
Autonomous systems of first-order ODEs. Trajectory and phase portrait. Equilibrium and its stability. Linearization.
Planar linear systems of ODEs with a constant regular matrix. Classification of equilibria.
Planar autonomous non-linear systems of ODEs. Topological equivalence.
Second-order autonomous non-linear equations.
Mathematical modeling in mechanics and biology. 
    Exercise

Analytical methods of solving systems of first order ODEs.
Analytical methods of solving higher-order ODEs.
Stability of linear systems of ODEs with constant coefficients.
Autonomous systems of first-order ODEs.
Planar linear systems of ODEs with a constant regular matrix - stability and classification of equilibria.
Planar autonomous non-linear systems of ODEs - stability and classification of equilibria.
Autonomous non-linear second-order equations - stability and classification of equilibria.
Mathematical modeling in mechanics and biology. 
Literature - fundamental:
1. W. E. Boyce, R. C. DiPrima, Elementary Differential Equations, 9th Edition, Wiley, 2008.
2. L. Perko, Differential Equations and Dynamical Systems, Text in Applied Mathematics, 7, Springer-Verlag, New York, 2001.
3. A. A. Andronov, E. A. Leontovich, I. I. Gordon, A. G. Maier, Qualitative Theory of Second-Order Dynamical Systems, John Wiley and Sons, New York, 1973.
Literature - recommended:
1. W. E. Boyce, R. C. DiPrima, Elementary Differential Equations, 9th Edition, Wiley, 2008.
2. L. Perko, Differential Equations and Dynamical Systems, Text in Applied Mathematics, 7, Springer-Verlag, New York, 2001.
3. J. Stewart, Calculus, 7th Edition, Cengage Learning, 2012.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
N-ENG-A full-time study --- no specialisation -- Cr,Ex 7 Compulsory 2 1 W