Mathematics III.
Code  Completion  Credits  Range  Language 

E011009  Z,ZK  5  2P+2C  English 
 Course guarantor:
 Stanislav Kračmar
 Lecturer:
 Stanislav Kračmar, Olga Majlingová
 Tutor:
 Stanislav Kračmar, Olga Majlingová
 Supervisor:
 Department of Technical Mathematics
 Synopsis:

An introductory course in ordinary differential equation and infinite series.
 Requirements:
 Syllabus of lectures:

Ordinary differential equations. Basic notions. Firstorder equations. Secondorder linear equations. Systems of equations in normal form. Autonomous systems. Linear systems. Linear systems with constant coefficients. Infinite series. Function series. Power series, Fourier series.
 Syllabus of tutorials:

Ordinary differential equations. Basic notions. Firstorder equations. Secondorder linear equations. Systems of equations in normal form. Autonomous systems. Linear systems. Linear systems with constant coefficients. Infinite series. Function series. Power series, Fourier series.
 Study Objective:

1. Ordinary differential equations of first order. Basic concepts. Maximal solution. Existence and uniqueness of maximal solution of the initial value problem., 2. Separable differential equations. Homogeneous differential equations of first order. Exact equation. Linear differential equation of first order. Bernoulli equation., 3. Systems of differential equations in normal form. Fundamental set of solutions of homogeneous linear systems. The Wronskian., 4. Linear differential equations of 2nd order. Method of undetermined coefficients., 5. Autonomous systems. Dynamic interpretation in the phase space., 6. Homogeneous linear autonomous systems. The Euler method for the general solution., 7. Phase diagram of the homogeneous linear autonomous system in the plane. Various types of equilibrium points. Nonhomogeneous linear autonomous systems., 8. Nonlinear autonomous systems. Properties of phase trajectories. First integral., 9. Infinite series of numbers. Tests of convergence for the series with positive terms., 10. Series with arbitrary real terms. Absolute and conditional convergence. The Leibnitz test., 11. Power series. Structure of the domain of convergence and determination of the domain., 12. Operations on power series (multiplication, differentiation, and integration of power series)., 13. The expansion of a function into the Taylor/MacLaurin series., 14. Application of power series to the solution of the initial value problem for the linear differential equation of 2nd order with variable coefficients.
 Study materials:

1. Burda, P.: Mathematics III, Ordinary Differential Equations and Infinite Series, CTU Publishing House, Prague, 1998.
 Note:
 Further information:
 No timetable has been prepared for this course
 The course is a part of the following study plans:

 12 74 79 00 BTZSI 2012 A  prezenční anglicky (compulsory course in the program)
 09 54 59 00 BSTR 2012 A  prezenční anglicky (compulsory course in the program)
 09 54 59 00 DSTR 2012 A  prezenční anglicky (compulsory course in the program)