1. SylabUZ
2. Wydział Nauk Ścisłych i Przyrodniczych
3. semestr zimowy 2024/2025
4. WMIiE - oferta ERASMUS - Program Erasmus
5. Differential Equations

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Differential Equations - opis przedmiotu

Cel przedmiotu

The main aim of this course is to familiarize students with the basic theory of ordinary differential equations: finding solutions of first-order and second-order ODE as well as first-order systems of ODE, the existence and the uniqueness of solutions of ODE, testing stability of critical points and making phase portraits of linear system in the plane.

Wymagania wstępne

Zapisz zmiany

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

Zakres tematyczny

Lecture

1. Basic concepts: n-order ordinary differential equation, system of n-order ordinary differential equations, solution of ODE, continuation of solution, saturated solutions, general and particular solution, integral curves, first integrals, phase space.
2. First-order ordinary differential equations. Examples of phenomena leading to ODE. Geometrical interpretation of ODE. Separable equations and the types of equations reducible to separable equations. Linear ODE and equations reducible to linear equations (Bernoulli equations, Riccati equations). Exact equations.
3.  Existence and uniqueness of local solutions of initial problems for ODE. Cauchy problem for ODE. Picard-Lindelöf theorem. Gronwall's lemma. 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.
4. Second-order ordinary differential equations. Physical motivation. Types of equations reducible to first-order ordinary differential equations. Linear second-order differential equations.
5. Systems of linear first-order differential equations. Existence and uniqueness of solutions. Systems of homogeneous equations, fundamental matrix. Systems of homogeneous equations with constant coefficients. Systems of inhomogenous equations.
6. Elements of the qualitative theory of ODE.

 

Class

1. Solving first-order ordinary differential equations: separable equations and the types of equations reducible to separable equations, linear equations, Bernoulli equations, Riccati equations, exact equations. Solving exercises related to physical phenomena which should be described in terms of Cauchy problems for ODE.
2. Solving exercises with the use of existence and uniqueness theorems of local solutions of initial problems for ODE.
3. Solving second-order ordinary differential equations by reducing them to first-order ordinary differential equations. Solving second-order linear ordinary differential equations.
4. Solving systems of first-order linear ordinary differential equations - computation of fundamental matrix.
5. Examination of stability of critical points of systems of first-order linear ODE. Sketching phase portraits.
    

Metody kształcenia

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

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

Class: learning outcomes will be verified through homeworks and test 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 classes (50%) 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.

Literatura podstawowa

1.  Ph. Hartman, Ordinary Differential Equations, Wiley, New York, 1964.
2. V. I. Arnold, Ordinary Differential Equations, Springer, 1992.
3. William E. Boyce, Richard C. DiPrima, Elementary differential equations and boundary value problems, Wiley, New York 2001.
4.  N. M. Matwiejew, Metody całkowania równań różniczkowych zwyczajnych, PWN, Warszawa 1970.
5.  N. M. Matwiejew, Zbiór zadań z równań różniczkowych zwyczajnych, PWN, Warszawa 1974

Literatura uzupełniająca

1. Marian Gewert, Zbigniew Skoczylas, Równania różniczkowe zwyczajne. Teoria, przykłady, zadania, Oficyna Wydawnicza GiS, Wrocław 2008.
2. Andrzej Palczewski, Równania różniczkowe zwyczajne, WNT, Warszawa 1999.
3. M. Braun, Differential Equations and Their Applications, An Introduction to Applied Mathematics, Springer, New York, 1983.

Uwagi

Zmodyfikowane przez dr Dorota Głazowska (ostatnia modyfikacja: 18-04-2024 13:08)