Mathematical Analysis (FSI-UMA-A)

Aim of the course: The aim of the course is to acquaint the students with the basic notions and methods of solving ordinary differential equations, with the fundamentals of the theory of stability of solutions to autonomous systems, and with other selected topics from the theory of ordinary differential equations. The task is also to show that the knowledge of the theory of ordinary differential equations can frequently be utilised in physics, technical mechanics, and other branches.

Acquired knowledge and skills: Students will acquire skills for analytical solution of higher order ordinary differential equations and systems of first order ordinary differential equations. They will be able to examine the stability of the equilibria (singular points) of non-linear autonomous systems. Students will also be enlightened on ordinary differential equations as mathematical models and on the qualitative analysis of the obtained equations.

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 70 points), discussion about the examinational test (maximum 10 points), and the evaluation of the oral part (maximum 20 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).

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 - methods of undetermined coefficients and variation of parameters.
Higher-order ordinary differential equations, solving higher-order linear ODEs with constant coefficients.
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.
Hamiltonian systems and second-order autonomous non-linear equations.
Mathematical modeling in mechanics and biology. 

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.