1. SylabUZ
2. Faculty of Mathematics, Computer Science and Econometrics
3. winter term 2020/2021
4. Mathematics - Second-cycle studies leading to MS degree
5. Differential Equations

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Differential Equations - course description

Aim of the course

The main aim of this course is to familiarize students with the theory of ordinary differential equations, with particular emphasis on the qualitative theory.

Prerequisites

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Mathematical Analysis 1 and 2, Linear Algebra 1 and 2, Mathematical Software.

Scope

1. First-order ordinary differential equations.
Basic concepts. Geometrical interpretation of ODE. ODE integrable by quadratures.
2. Existence and uniqueness of local solutions of the initial problems for ODE.
Cauchy problem for ODE. Existential theorems (Picard-Lindelöf theorem, Peano theorem). Extension of solutions of the initial problems for ODE. Dependence of the solution to Cauchy problem on initial conditions and the right-hand side of the equation.
3. High-order ordinary differential equations.
Types of equations reducible to first-order ordinary differential equations. Linear second-order differential equations. Sturm-Liouville boundary problem.
4. Dynamical interpretation of systems of ODE.
Autonomous systems. Phase trajectories and phase portraits. Flows and orbits. First integrals.
5. Systems of linear ordinary differential equations.
Methods of solving of homogeneous and inhomogeneous systems of linear equations. Classification and stability of critical points of systems of linear ODE in the plane. Phase portraits.
6. Systems of nonlinear ordinary differential equations.
Local phase portraits. Linearization, Grobman-Hartman theorem. Classification and stability of critical points of systems of nonlinear ODE in the plane. Global phase portraits.

7. Periodic orbits and limits cycles.
Limits sets. Poincaré-Bendixson theorem.
8. Elements of the stability theory.
Lyapunov stability. Hurwitz theorem. Lyapunov function and fundamental stability theorems.
9. Bifurcations and chaos.
Hopf bifurcation. The Lorenz model.
10. Some differential models in physics, biology, medicine and economics.
Van der Pol oscillator. Lotka-Volterra systems.

Teaching methods

Classes. Solving of problems related to contents of lectures with particular emphasis on practical applications of learned concepts.
Laboratory. Solving of problems related to ODE by means of mathematical software.

Traditional lectures; classes with the lists of exercises to solve by students; computer lab.

Learning outcomes and methods of theirs verification

Outcome description	Outcome symbols	Methods of verification	The class form

Assignment conditions

Class and Laboratory: learning outcomes will be verified through two tests consisted of exercises of different degree of difficulty. A grade determined by the sum of points from these two tests is a basis of assessment.
Lecture: final exam. A grade determined by the sum of points from that exam is a basis of assessment.
A grade from the course is consisted of the grade from laboratory (20%), the grade from classes (30%) and the grade from the final exam (50%). To take a final exam, students must receive a positive grade from classes. To attain a pass in the course students are required to pass the final exam.

Recommended reading

1. A. Palczewski, Równania różniczkowe zwyczajne, WNT, Warszawa, 1999.
2. W. I. Arnold, Równania różniczkowe zwyczajne, PWN, Warszawa, 1975.
3. D. K. Arrowsmith, C.M. Place, Ordinary differential equations, A qualitative approach with applications, Chapman and Hall, London, 1982.
4. A. Pelczar, J. Szarski, Wstęp do równań różniczkowych zwyczajnych, PWN, Warszawa, 1987.
5. N. M. Matwiejew, Metody całkowania równań różniczkowych zwyczajnych, PWN, Warszawa, 1986.

Further reading

1. L. S. Pontriagin, Równania różniczkowe zwyczajne, PWN, Warszawa, 1964.
2. Ph. Hartman, Ordinary Differential Equations, Wiley, New York, 1964.

Notes

Modified by dr Alina Szelecka (last modification: 18-09-2020 13:46)