Mathematics III.
Code  Completion  Credits  Range  Language 

E011093  Z,ZK  4  2P+2C+0L  English 
 Garant předmětu:
 Lecturer:
 Tutor:
 Supervisor:
 Department of Technical Mathematics
 Synopsis:

Infinite series. Number series. Convergence criteria for series with nonnegative terms.
Absolute and relative convergence. Alternating series, Leibniz's criterion.
Series of functions, the domain of convergence. Power series. Center and radius of convergence. Examination of the interval of convergence and the domain of convergence.
Operations with the power series. Expansion of functions into Taylor series.
Fourier series. Calculation of Fourier coefficients, the convergence of Fourier series.
Approximation of functions by trigonometric polynomials. Cosine and sine Fourier series.
Ordinary differential equations. Firstorder equations.
Sufficient conditions for the existence and uniqueness of the maximal solution of the Cauchy’s problem.
Secondorder linear equations. The structure of the set of solutions. The fundamental system, the general solutions, particular solutions. Physical interpretation.
Systems of equations in the normal form. Autonomous systems. Equilibrium points, trajectories of systems.
Linear systems. The fundamental system, general solutions, particular solutions.
Linear systems with constant coefficients. Euler's method. Solution of nonhomogeneous systems.
Elimination method. Solution of differential equations using power series.
 Requirements:
 Syllabus of lectures:

Infinite series. Number series. Convergence criteria for series with nonnegative terms.
Absolute and relative convergence. Alternating series, Leibniz's criterion.
Series of functions, the domain of convergence. Power series. Center and radius of convergence. Examination of the interval of convergence and the domain of convergence.
Operations with the power series. Expansion of functions into Taylor series.
Fourier series. Calculation of Fourier coefficients, the convergence of Fourier series.
Approximation of functions by trigonometric polynomials. Cosine and sine Fourier series.
Ordinary differential equations. Firstorder equations.
Sufficient conditions for the existence and uniqueness of the maximal solution of the Cauchy’s problem.
Secondorder linear equations. The structure of the set of solutions. The fundamental system, the general solutions, particular solutions. Physical interpretation.
Systems of equations in the normal form. Autonomous systems. Equilibrium points, trajectories of systems.
Linear systems. The fundamental system, general solutions, particular solutions.
Linear systems with constant coefficients. Euler's method. Solution of nonhomogeneous systems.
Elimination method. Solution of differential equations using power series.
 Syllabus of tutorials:
 Study Objective:
 Study materials:

Burda, P.: Mathematics III, Ordinary Differential Equations and Infinite Series, CTU Publishing House, Prague, 1998.
Robinson, James C.: An Introduction to Ordinary Differential Equations, Cambridge University Press 2004
ISBN: ISBN number:9780521826501, ISBN 9780511164033
 Note:
 Further information:
 No timetable has been prepared for this course
 The course is a part of the following study plans: