Ordinary differential and difference eguations.

obligatory courses

(in Polish) Założenia wstępne: Student powinien posiadać umiejętność posługiwania się rachunkiem różniczkowym i całkowym oraz pojęciami algebry liniowej w zakresie kursu tych przedmiotów dla pierwszego roku WNE lub jakiegokolwiek wydziału o kierunku matematycznym, przyrodniczym lub technicznym.

Theory of differential equations gives universal tools to describe dynamical processes which are objects of research in many domains of science, including economics and sociology. One of the first application of differential equations to economics was presented by M.Kalecki in 1936. Today, most of the models in economic sciences are based on differential equations. The aim of the Seminar is to learn basic types of the equations and basic methods of solution, and some applications.

The basic notions of the theory of differential equations are derivative and differential which were introduced by G.W.Leibniz and I.Newton in 17th century. Since then we observe a fast progress in many domains of science. It seems that whenever we try to describe a dynamical process (evolution of a system) we have to use differential equations. The program of the Seminar is the following:

1. basic differential equations: equations with separated variables, first order linear equations, Bernoulli’s and Riccati’s equations, second order linear equations;

2.systems of linear equations with constant coefficients;

3. nonlinear systems of first order differential equations, critical points, phase portraits: introduction to qualitative theory of ordinary differential equation;

4. Lagrange and Hamilton equations;

5. difference equations and numerical solutions.

A.Palczewski, Równania różniczkowe zwyczajne, WNT 2004;

H.Amann, Ordinary Differential Equations, Gruyter, Berlin 1990;

P.Hartman, Ordinary Differential Equations, J.Wiley and Sons, N.Y. 1964;

D.K.Arrowsmith, C.M.Place, Ordinary Differential Equations, A Qualitative Approach with Applications, Chapman and Hall, London 1982

Student knows basic types of ordinary differential equations, the basic methods of solutions of the equations and its systems, some examples of its applications to economic sciences and sociology.

Exam