Course detail

# Mathematics III

The course provides an introduction to the theory of infinite series and the theory of ordinary and partial differential equations. These branches form the theoretical background in the study of many physical and engineering problems. The course deals with the following topics:
Number series. Function series. Power series. Taylor series. Fourier series.
Ordinary differential equations. First order differential equations. Higher order linear differential equations. Systems of first order linear differential equations.
Partial differential equations. Classification.

Language of instruction

Czech

Number of ECTS credits

8

Mode of study

Not applicable.

Learning outcomes of the course unit

Students will acquire knowledge of basic types of differential equations. They will be made familiar with differential equations as mathematical models of given problems, with problems of the existence and uniqueness of the solution and with the choice of a suitable solving method. They will master solving of problems of the convergence of infinite series as well as expansions of functions into Taylor and Fourier series.

Prerequisites

Linear algebra, differential and integral calculus of functions in a single and more variables.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises 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: Active participation in seminars. Fulfilment of all conditions of the running control of knowledge. At least half of all possible 20 points in both check tests (the first test takes place in 8th week of the semester, the second one in 13th week of the semester). If a student does not fulfil this condition, the teacher can set an alternative one.

Examination: The examination tests the knowledge of definitions and theorems (especially the ability of their application to the given problems) and practical skills in solving of examples. The exam is written (possibly followed by an oral part). The written exam consists in particular of the examples on the following topics:
Number and function series, the expansion of a function into Taylor series, solving of first order ODEs, solving of higher order linear ODEs, solving of system of first order linear ODEs, Fourier series, solving of ODEs via the infinite series and the Laplace tranform method, boundary value problems, basics of PDEs theory. Some theoretical queries concerning basic concepts are included in the written part as well.
Topics of practical part: The expansion of a function into Taylor series, solving of first order ODEs, solving of higher order linear ODEs, solving of system of first order linear ODEs.
The final grade reflects the result of the written part of the exam (maximum 75 points), the results achieved in seminars (maximum 20 points) and the results achieved in seminars in computer labs (maximum 5 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

Not applicable.

Work placements

Not applicable.

Aims

The aim of the course is to explain basic notions and methods of solving ordinary and partial differential equations, and foundations of infinite series theory. The task of the course is to show that knowledge of the theory of differential equations can be utilized especially in physics and technical branches. Moreover, it is shown that foundations of infinite series theory are important tools for solving various problems.

Specification of controlled education, way of implementation and compensation for absences

Attendance at lectures is recommended, attendance at seminars is obligatory and checked. Lessons are planned according to the week schedules. Absence from seminars may be compensated for by the agreement with the teacher.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Hartman, P.: Ordinary Differential Equations, New York, 1964. (EN)

Čermák, J., Nechvátal, L.: Matematika III, Brno, 2016. (CS)
Logan, J.D.: A First Course in Differential Equations. New York, Springer, 2006. (EN)
Čermák, J.: Sbírka příkladů z Matematické analýzy III a IV, Brno, 1998. (CS)

Classification of course in study plans

• Programme B3A-P Bachelor's

branch B-MTI , 2. year of study, winter semester, compulsory
branch B-MET , 2. year of study, winter semester, compulsory

• Programme B3S-P Bachelor's

branch B-STI , 2. year of study, winter semester, compulsory
branch B-KSB , 2. year of study, winter semester, compulsory

#### Type of course unit

Lecture

39 hours, optionally

Teacher / Lecturer

Syllabus

1. Number series. Basic notions. Convergence criteria.
2. Operations with number series. Function series. Basic properties.
3. Power series. Taylor series and expansions of functions into power series.
4. Trigonometric Fourier series. Problems of the convergence and expansions of functions.
5. Ordinary differential equations (ODE). Basic notions. The existence and uniqueness of the solution to the initial value problem for 1st order ODE. Analytical methods of solving of 1st order ODE.
6. Higher order ODEs. Basic notions. The existence and uniqueness of the solution to the initial value problem for higher order ODEs. Genral solutions of homogeneous and nonhomogeneous linear equations. Methods of solving of higher order homogeneous linear ODEs with constant coefficients.
7. Methods of solving of higher order non-homogeneous linear ODEs with constant coefficients.
8. Systems of 1st order ODEs. Basic notions. The existence and uniqueness of the solution to the initial value problem for systems of 1st order ODE. General solution of homogeneous and non-homogeneous systems of 1st order ODE.
9. Methods of solving of homogeneous systems of 1st order linear ODEs.
10. Methods of solving of non-homogeneous systems of 1st order linear ODEs.
11. The Laplace transform and its use in solving of linear ODEs. The method of Taylor series in solving of ODEs.
12. Stability of solutions of ODEs and their systems. Boundary value problem for 2nd order ODEs. Partial differential equations. Basic notions. The equations of mathematical physics.
13. Mathematical modelling by differential equations.

Exercise

39 hours, compulsory

Teacher / Lecturer

Syllabus

1. Limits and integrals - revision.
2. Infinite series.
3. Function and power series.
4. Taylor series.
5. Fourier series.
6. Analytical methods of solving of 1st order ODEs.
7. Analytical methods of solving of 1st order ODEs (continuation).
8. Higher order linear homogeneous ODEs.
9. Higher order non-homogeneous linear ODEs.
10. Laplace transform method of solving of linear ODEs.
11. Systems of 1st order linear homogeneous ODEs.
12. Systems of 1st order linear non-homogeneous ODEs.
13. Fourier method of solving of PDEs.

Computer-assisted exercise

13 hours, compulsory

Teacher / Lecturer

Syllabus

The course is realized in computer labs. The MAPLE software is utilized to illustrate and complete the following topics: 1. Revision of basic skills in MAPLE. 2. Function series - graphical illustrations of types of the convergence (with a special emphasize on Taylor and Fourier series). 3. 1st order ODEs - geometrical interpretation of solutions, graphical methods of solving (direction fields). 4. 1st order ODEs - applications (orthogonal trajectories and others). 5. Higher order ODEs - graphical interpretations of solutions, Taylor series method. 6. Systems of 1st order ODEs - graphical interpretations of solutions, the phase portrait. 7. PDEs - selected methods of solving.