Předmět Ordinary differential equations (KMA / WOBDR)

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Materiál	Typ	Datum	Počet stažení

Obsah

Programme of the course1.Introduction to ordinary differential equations.2.Cauchy initial valued problem for ODE, Theorem on the existence and uniqueness of the solution of the initial valued problem, Piccard's approximations, Euler's polygons, elementary methods of integration, extensions of solutions.3.Geometric interpretation of ODEs of the first order: linear element, direction fields (slope fields), isoclines, integral curves.4.Basic kinds of ODEs of the first order and methods of its solutions, other kinds of ODEs of the first order: Ricatti equations, exact differential equations, integrating factor.5.Implicit differential equations: singular solution, Lagrange equations, Clairaut's equations. Classification of ordinary differential equations.6.Implicit differential equations: singular solution, Lagrange equations, Clairaut's equations. Classification of ordinary differential equations.7.Differential equations of the higher order. Some basic kinds of equations of the higher order. Reduction of order of a differential equation using first integrals or using some substitutions.8.Linear differential equations of the n-th order with constant coefficients homogeneous and non-homogeneous. Methods of solution of homogeneous equations of the n-th order: characteristic equation. Methods of solution of non-homegeneous equations of the n-th order: method of variation of constants, method of undetermined coefficients for non-homogeneous equations with a special right-hand side. Euler differential equations of the n-th order.9.Systems of ordinary differential equations. Physical motivation: Coupled oscillation. Systems of ODEs in normal form (autonomous systems). Geometric interpretation of the solution of the autonomous system.10.Existence and uniqueness of the solution of the autonomous system of ordinary differential equations. Methods of solution of systems of ODEs: elimination method, first integrals.11.Autonomous linear systems and their general solution, phase space of an autonomous system, trajectory (phase picture), limit sets, stationary solution, limit cycle. Trajectory (phase picture) of a linear two-dimensional autonomous system, knot, saddle, focus, centre.12.Systems of linear differential equations with constant coefficients, homogeneous, non-homogeneous. The methods of solution of homogeneous systems of linear differential equations with constant coefficients: Euler's method, elimination method.13.Systems of linear differential equations with constant coefficients, homogeneous, non-homogeneous. The methods of solution of homogeneous systems of linear differential equations with constant coefficients: Euler's method, elimination method. Methods of solution of non-homogeneous systems of linear differential equations with constant coefficients: method of variation constants.14.Ordinary differential equations (ODEs) as simple models of real processes: mechanical motions, harmonic oscillator, damped oscillation, forced oscillation, resonance; cooling of a body, radioactive decay, reproduction, blending of mixtures, applications in chemistry, in biology, in economics.

Literatura

ZILL, D. G. A First Course in Differential Equations with Applications. TIERNEY, J. A. Differential Equations. Boston, London, Sydney, Toronto: Allyn and Bacon, Inc., 1985. EDWARDS, C. H.; PENNEY, D. E. Elementary Differential Equations with Applications. HARTMAN, P. Ordinary Differential Equations. John Wiley & Sons, 1973.

Požadavky

Requirement during the semester, before the examination: to solve all problems within individual exercises. The examination consists of a written test and an oral interview. The written part will be evaluated by scoring. The maximum score is 100 points.The evaluation of the course including the classification is carried out in accordance with the Study and Examination Regulations OU.

Garant

RNDr. Martin Swaczyna, Ph.D.

Vyučující

RNDr. Martin Swaczyna, Ph.D.RNDr. Martin Swaczyna, Ph.D.