Mathematics III (FSI-3M-A)

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 plays an important role in physics and technical branches. Moreover, it is shown that the infinite series theory is a necessary tool for solving various problems.
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 problems of the convergence of infinite series as well as expansions of functions into Taylor and Fourier series.

The course provides an introduction to the theory of ordinary differential equations an the theory of infinite series. These branches form the theoretical background in the study of many physical and engineering problems. The course deals with the following topics:
Ordinary differential equations. First order differential equations. Higher order linear differential equations. Systems of first order linear differential equations. Partial differential equations. Modelling with differential equations.
Basic numerical methods for solving differential equations with a suitable software (e.g. Matlab).
Number series. Function series. Power series. Taylor series. Fourier series.

Course-unit credit is awarded on the following conditions: Active participation in seminars fulfilment of all conditions of the running control of knowledge (this concerns also the seminars in computer lab). At least half of all possible points in each of the two tests should be obtained.

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 has written and oral part. The written exam consists in particular of the examples on the following topics:
Solving first order ODEs, solving higher order linear ODEs, solving system of first order linear ODEs, Fourier series, solving ODEs via the infinite series and the Laplace transform method, boundary value problems, basics of PDEs theory,
number and function series, application of convergence tests, the expansion of a function into Taylor series and manipulations with this expansion.
Some theoretical questions concerning basic concepts can be included in the written part as well.
The final grade reflects the result of the written part of the exam (maximum 80 points) and the result of the oral part (maximum 20 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).

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.

- Ordinary differential equations (ODE). Basic notions. The existence and uniqueness of the solution to the initial value problem.
- Analytical methods of solving of 1st order ODEs.
- Higher order ODEs. Basic notions. The existence and uniqueness of the solution to the initial value problem. The structure of general solutions.
- Methods of solving higher order homogeneous and nonhomogeneous linear ODEs with constant coefficients.
- Systems of 1st order ODEs. Basic notions. The existence and uniqueness of the solution to the initial value problem. The structure of general solutions.
- Methods of solving homogeneous and nonhomogeneous systems of 1st order linear ODEs.
- The Laplace transform and its use in solving of linear ODEs.
- Stability. Analysis in the phase plane.
- Boundary value problem for 2nd order ODEs.
- Partial differential equations. Basic notions. The equations of mathematical physics.
- Mathematical modelling by differential equations.
- Number series. Basic notions. Convergence criteria. Operations with number series.
- Function series. Basic properties.
- Power series. Taylor series, expansions of functions into power series, and applications.
- Trigonometric Fourier series. Problems of the convergence and expansions of functions.

- Limits and integrals - revision.
- Analytical methods of solving 1st order ODEs.
- Higher order linear homogeneous and nonhomogeneous ODEs.
- Systems of 1st order linear homogeneous and nonhomogeneous ODEs.
- Laplace transform method and series method of solving of ODEs.
- Boundary value problems and elements of PDEs.
- Infinite series. Convergence tests.
- Function and power series.
- Taylor series.
- Fourier series.

Boyce, W. E., DiPrima, R. C., Elementary Differential Equations, 9th Edition, Wiley, 2008. (EN)