FSI-CM-KAcad. year: 2019/2020
The course is intended as an introduction to basic methods applied for solving of ordinary differential equations and problems of mathematical statistics.
The knowledge of the basic theory of differential equations and methods of solving is an important foundation for further study of physical and technical disciplines, especially those connected with mechanics.
Statistical methods are concentrated on descriptive statistics, random events, probability, random variables and vectors, random sample, parameters estimation and tests of hypotheses. The practicals cover problems and applications in mechanical engineering.
Learning outcomes of the course unit
Students obtain necessary knowledge of ordinary differential equations and mathematical statistics, which enables them to understand and apply deterministic and stochastic models of technical phenomenon based on these methods.
Foundations of differential and integral calculus.
Recommended optional programme components
Recommended or required reading
Čermák,J.- Ženíšek, A.: Matematika III. Brno: FSI VUT V Akademickém nakladatelství CERM Brno, 2001
Hartman, P.: Ordinary Differential Equations. New York: John Wiley & Sons, 1964.
Sprinthall, R. C.: Basic Statistical Analysis. Boston : Allyn and Bacon, 1997.
Karpíšek, Z.: Matematika IV - Statistika a pravděpodobnost. 2. vydání. Brno : FSI VUT v Akademickém nakladatelství CERM Brno, 2003.
Montgomery, D. C. - Renger, G.: Probability and Statistics. New York : John Wiley & Sons, Inc.,1996.
Karpíšek, Z. – Popela, P. – Bednář, J.: Statistika a pravděpodobnost. Učební pomůcka - studijní opora pro kombinované studium. Brno : FSI VUT v Akademickém nakladatelství CERM Brno, 2002.
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 attendance at seminars, understanding of the subject-matter. Fulfilment of all conditions of the running control of knowledge. At least half of all possible 30 points in both check tests. If a student does not fulfil this condition, the teacher can set an alternative one.
Examination (written form): practical part (2 examples from ordinary differential equations; 2 examples from mathematical statistics) with own summary of formulas; theoretical part (4 questions concerning basic terms, their properties, sense and practical use);
Examination (evaluation): The final grade reflects the result of the written part of the exam (maximum 70 points) and the results achieved in seminars (maximum 30 points); classification according to the total sum of points achieved: 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).
Language of instruction
The aim is to acquaint students with basic terms and methods of solving of ordinary differential equations and mathematical statistics. Another goal of the course is to form the student's thinking in modelling of real phenomenon and processes in engineering fields.
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.
Type of course unit
13 hours, optionally
Teacher / Lecturer
1. ODE. Basic terms. Existence and uniqueness of solutions.
2. Analytical methods of solving of 1st order ODE.
3. Higher order ODEs. Properties of solutions and methods of solving of higher order homogeneous linear ODEs.
4. Properties of solutions and methods of solving of higher order non-homogeneous linear ODEs.
5. Systems of 1st order ODEs. Properties of solutions and methods of solving of homogeneous linear systems of 1st order ODEs.
6. Properties of solutions and methods of solving of non-homogeneous linear systems of 1st order ODEs.
7. Boundary value problem for 2nd order ODEs.
8. Descriptive statistics.
9. Random events and probability.
10. Random variable and vector, functional and numerical characteristics.
11. Basic probability distributions (Bi, H, Po, N), properties and use.
12. Random sample, parameter estimations (Bi, N).
13. Testing statistical hypotheses of parameters (Bi, N).
26 hours, compulsory
Teacher / Lecturer
1. Calculation of integrals - revision.
2. Analytical methods of solving of 1st order ODEs.
3. Analytical methods of solving of 1st order ODEs (continuation).
4. Higher order linear homogeneous ODEs.
5. Higher order non-homogeneous linear ODEs.
6. Systems of 1st order linear homogeneous ODEs.
7. Systems of 1st order linear non-homogeneous ODEs.
8. Descriptive statistics (univariate and bivariate sample).
9. Probability, conditioned probability, independent events.
10. Functional and numerical characteristics of random variable.
11. Probability distributions (Bi, H, Po, N).
12. Point and interval estimates of parameters N and Bi.
13. Testing hypotheses of parameters N and Bi.