Ordinary differential and difference eguations. 2400-M1IiERR
The beginnings of differential calculus and its applications. Ideas of Leibniz and Newton: principles of optimal action (Leibniz) and laws of dynamics of systems (Newton). First applications of differential equations in natural sciences and sociology (models of population development). Basic types of ordinary differential equations: equations with separable variables, homogeneous, linear equations of the first and second order, Bernoulli and Riccati equations, systems of first order equations and their phase images. Macroeconomic model of M. Kalecki – logistic curves. linearization of nonlinear systems, Lotka-Voltera system. Elements of qualitative theory of differential equations: classification of stationary points and equivalence of phase images in the plane. Extremal problems, Euler-Lagrange and Hamilton equations. Difference equations, single- and multi-step difference schemes, examples.
Type of course
Prerequisites (description)
Course coordinators
Learning outcomes
After completing the course, the student is able to use the most commonly used types of differential equations and systems of equations in practice, is able to solve them and analyze the properties of solutions. Is able to build simple differential models of some dynamic processes and read with understanding scientific papers in which such models are created and analyzed. Also has knowledge of some approximate methods of solving ordinary differential equations and the ability to use some computer programs for such approximate calculations. Also has some knowledge of extremal issues (optimization) and is prepared to expand it by reading with understanding the appropriate literature on the subject.
Assessment criteria
During classes, the student is expected to be active, ready to solve simpler tasks at the board and prepare presentations. The final work is assessed in the form of a series of tasks on tests and possibly the development of examples of applications of differential equations in social sciences.
Evaluation: 75% of points for three tasks and 25% for activity.
Bibliography
Andrzej Palczewski, Równania różniczkowe zwyczajne, teoria i metody numeryczne , WNT 2004
D.K.Arrowsmith, C.M.Place, Ordinary Differential Equations, London, New York 1982,
R.Haberman, Mathematical Models, Prentice-Hall, 1977
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
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