Czech

Not applicable.

The ability to orientate in the basic notions and problems
of differential equations.
Solving problems in the areas cited in the annotation above
by use of these methods. Solving these problems by use of
modern mathematical software.

The subject knowledge on the Bachelor´s degree level is requested.

Not applicable.

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations.

Abilities leading to successful solution of some classes of differential equations as well as necessary theoretical knowledge will be positively estimated.

I. Differential equations of the first order. Basic notions. Existence of solutions. Successive approximations. A summary of basic classes of analytically solvable differential equations of the first order. Higher-order equations. Solution of linear equations of the second-order with power series. Bessel’s equation and Bessel‘s functions.
II. Existence and unicity of solutions of systems differential equations of the first order. Linear systems of ordinary differential equations. General properties of solutions and the structure of family of all solutions. The transient matrix. Solving of initial problem with transient matrix. Linears systems with constant coefficients (homogeneous systems – eliminative method, method of characteristic values, application of the matrix exponential, Putzer’s algorithm, nonhomogeneous systems – method of undetermined coefficients, method of variation of constants). Characterization of circuits by linear systems.
III. Stability of solutions of systems of differential equations. Autonomous systems. Lyapunov direct method for autonomous systems. Lyapunov‘ functions. Lyapunov direct method for nonautonomous systems. Stability of linear systems. Hurwitz‘s criterion. Michailov‘s criterion. Stability by linear approximation. Phase analysis of linear two-dimensional autonomous system with constant coefficients, cases of stability.
IV. Limit cycles and periodic solutions. Poincaré-Bendixson’s theorem. Liénard-Levinson-Smith’s theorem. Aplication to nonlinear equations describing periodic state of electrical processes.
IV. Partial differential equations of the first-order. Initial problem. Simplest classes of equations. Characteristic system. Existence of solutions. General solution. First integrals. Pfaff’s equation.
V. Partial differential equations of the second-order. Classification of equations. Transformatin of variables. Wave equation, D’Alembert’s formula. Heat equation, Dirichlet’s problem. Laplace‘s equation. Fourier’s method of separated variables.

Not applicable.

Differential equations are the base of many fields of engineering science. The purpose of this course is to develop the basic notions concerning the properties of solutions of differential equations and to give the basic techniques for solution of differential equations. In this course not only several exact solution methods are explained, but attention is focused also on possibilities for getting information concerning properties of solutions. Methods are illustrated on concrete electrical-engineering examples.

The content and forms of instruction in the evaluated course are specified by a regulation issued by the lecturer responsible for the course and updated for every academic year. Necessary conditions for course-unit credit are - regular attendance, nonzero assessment of half-semester written test and final written test.

Not applicable.

Not applicable.

Not applicable.

Not applicable.