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Course detail
FSI-UMA-AAcad. year: 2022/2023
The course provides an introduction to the theory of differential equations and dynamical systems. These branches form the theoretical background for mathematical modeling in physics, mechanics, and other disciplines.
Language of instruction
Number of ECTS credits
Mode of study
Guarantor
Department
Learning outcomes of the course unit
Students will acquire skills for analytical solving 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 be also enlightened on ordinary differential equations as mathematical models and on the qualitative analysis of the obtained equations.
Prerequisites
Linear algebra, differential calculus of functions of one and several variables, integral calculus of functions of one variable, first order ordinary differential equations.
Co-requisites
Planned learning activities and teaching methods
The course is taught through lectures explaining the basic principles and theory of the discipline. Seminars are focused on practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes
Course-unit credit is awarded on the following conditions: A semestral project consisting of the assigned problems; the work will be evaluated. Active participation in seminars (unless the student attends the course in the form of consultations).
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 examination has written and oral part.
The final grade reflects the evaluation of the semestral project (maximum 15 points), the result of the examinational test (maximum 75 points), discussion about the examinational test (maximum 10 points).
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).
Course curriculum
Work placements
Aims
The aim of the course is to acquaint the students with basic notions and methods of solving of 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.
Specification of controlled education, way of implementation and compensation for absences
Attendance at lectures is recommended, attendance at seminars is obligatory and checked. Absence from seminars may be compensated for by the agreement with the teacher.
Recommended optional programme components
Prerequisites and corequisites
Basic literature
Recommended reading
Classification of course in study plans
branch CZV , 1 year of study, winter semester, compulsory
Lecture
Teacher / Lecturer
Syllabus
1. First order ordinary differential equations (revision). Higher order ordinary differential equations (ODE). Basic notions. The existence and uniqueness of a solution to the initial value problem. General solutions of homogeneous and non-homogeneous linear equations.
2. Methods of solving of higher order homogeneous linear ODEs with constant coefficients. Solving of higher order non-homogeneous linear ODEs - methods of variation of parameters.
3. Solving of higher order non-homogeneous linear ODEs with constant coefficients - methods of undetermined coefficients.
4. Systems of first order ordinary differential equations. The existence and uniqueness of a solution to the initial value problem. General solutions of homogeneous and non-homogeneous linear systems.
5. Methods of solving of homogeneous systems of linear ODEs with constant coefficients.
6. Solving of non-homogeneous systems of linear ODEs with constant coefficients - methods of variation of parameters.
7. Solving of non-homogeneous systems of linear ODEs with constant coefficients - methods of undetermined coefficients.
8. Stability of solutions to ordinary differential equations and their systems. Basic notions. Stability of linear systems of ODEs with constant coefficients.
9. Autonomous systems of first order ODEs. Orbit and phase portrait. Equilibrium and its stability. Linearization.
10. Two-dimensional linear systems of ODEs with a constant regular matrix. Classification of equilibria.
11. Two-dimensional autonomous non-linear systems of ODEs. Topological equivalence.
12. Mathematical modeling in mechanics and biology.
13. Other selected topics from the theory of ordinary differential equations.
Exercise
1. First order ordinary differential equations (revision). Analytical methods of solving of higher order ODEs.
2. Analytical methods of solving of higher order ODEs - continuation.
3. Analytical methods of solving of higher order ODEs - continuation.
4. Analytical methods of solving of systems of first order ODEs.
5. Analytical methods of solving of systems of first order ODEs - continuation.
6. Analytical methods of solving of systems of first order ODEs - continuation.
7. Analytical methods of solving of systems of first order ODEs - continuation.
8. Stability of linear systems of ODEs with constant coefficients.
9. Autonomous systems of first order ODEs.
10. Two-dimensional linear systems of ODEs with a constant regular matrix.11. Two-dimensional autonomous non-linear systems of ODEs.