Description:

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. First-order equations.
Sufficient conditions for the existence and uniqueness of the maximal solution of the Cauchy?s problem.
Second-order 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 non-homogeneous systems.
Elimination method. Solution of differential equations using power series.

Contents:

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. First-order equations.
Sufficient conditions for the existence and uniqueness of the maximal solution of the Cauchy?s problem.
Second-order 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 non-homogeneous systems.
Elimination method. Solution of differential equations using power series.

Recommended literature:

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
https://ebookcentral.proquest.com/lib/cvut/detail.action?docID=255188&query=ordinary+differential+equations

Abbreviations used:

Semester:

• W ... winter semester (usually October - February)
• S ... spring semester (usually March - June)
• W,S ... both semesters

Mode of completion of the course:

• A ... Assessment (no grade is given to this course but credits are awarded. You will receive only P (Passed) of F (Failed) and number of credits)
• GA ... Graded Assessment (a grade is awarded for this course)
• EX ... Examination (a grade is awarded for this course)
• A, EX ... Examination (the award of Assessment is a precondition for taking the Examination in the given subject, a grade is awarded for this course)

Weekly load (hours per week):

• P ... lecture
• C ... seminar
• L ... laboratory
• R ... proseminar
• S ... seminar