# Theory of Differential Equations (MATH562) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Theory of Differential Equations MATH562 3 0 0 3 5
Pre-requisite Course(s)
Consent of the Department
Course Language English N/A Natural & Applied Sciences Master's Degree Face To Face Lecture, Question and Answer. The course aims to introduce and present Initial Value Problem: Existence and Uniqueness of Solutions; Continuation of Solutions; Continuous and Differential Dependence of Solutions. Linear Systems: Linear Homogeneous And Nonhomogeneous Systems with Constant and Variable Coefficients; Structure of Solutions of Systems with Constant and Periodic Coefficients; Higher Order Linear Differential Equations; Sturmian Theory, Stability: Lyapunov Stability and Instability. Lyapunov Functions; Lyapunov's Second Method; Quasilinear Systems. The students who succeeded in this course; students are expected to know and understand various ideas, concepts and methods from ordinary differential equations and how these ideas may be used in, or are connected to, the fields of engineering and mathematics. students will be able to learn how they can construct a Lyapunov function to determine the stability/instability of ordinary differential equations (ODE) and construct a new ODE to determine the oscillation/nonoscillation of an ODE. IVP: existence and uniqueness, continuation and continuous dependence of solutions; linear systems: linear (non)homogeneous systems with constant and variable coefficients; structure of solutions of systems with periodic coefficients; higher order linear differential equations; Sturmian theory, stability: Lyapunov (in)stability, Lyapunov functions

### Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 Initial Value Problem (IVP): Examples of (IVP) Read related sections in references
2 Fundamental Theory: Preliminaries, Existence and Uniqueness of Solutions; Read related sections in references
3 Continuation of Solutions; Continuity and Differentiability of Solutions with respect to parameters Read related sections in references
4 Linear Systems: Preliminaries, Linear Homogeneous and Nonhomogeneous Read related sections in references
5 Linear Systems with Constant and Variable Coefficients Read related sections in references
6 Structure of Solutions of Systems with Constant and Periodic Coefficients; Read related sections in references
7 Midterm
8 Higher Order Linear Differential Equations; Read related sections in references
9 Sturmian Theory: Sturm Comparison Theory, Sturm Oscillation Theory. Read related sections in references
10 Stability: Definitions of Stability and Boundedness. Read related sections in references
11 Lyapunov Stability and Instability Read related sections in references
12 Lyapunov Functions; Lyapunav stability and Instability results. Lyapunov's Second Method;. Read related sections in references
13 Quasilinear Systems; Linearization Read related sections in references
14 Stability of an Equilibrium and Stable Manifold Theorem for Nonautonomous Differential Equations Read related sections in references
15 Revision.
16 Midterm

### Sources

Course Book 1. Richard K. Miller, Anthony N. Michel, Ordinary Differential Equations, 1982, Academic Press 2. W. Kelley, A. Peterson, The Theory of Differential Equations Classical and Qualitative,2004, Prentice–Hall. 3. C. A. Swanson, Comparison and Oscillation Theory, 1968, Academic Press.

### Evaluation System

Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics - -
Homework Assignments 5 30
Presentation - -
Project - -
Report - -
Seminar - -
Midterms Exams/Midterms Jury 1 30
Final Exam/Final Jury 1 40
Toplam 7 100
 Percentage of Semester Work 60 40 100

### Course Category

Core Courses X

### The Relation Between Course Learning Competencies and Program Qualifications

# Program Qualifications / Competencies Level of Contribution
1 2 3 4 5
1 Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area.
2 Has the ability to obtain, to evaluate, to interpret and to apply information by doing scientific research.
3 Can apply gained knowledge and problem solving abilities in inter-disciplinary research.
4 Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary.
5 Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study.
6 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework.
7 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility.
8 Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation.
9 Has proficiency in English language and has the ability to communicate with colleagues and to follow the innovations in mathematics and related fields.
10 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge.
11 Has professional ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications.

Activities Number Duration (Hours) Total Workload
Course Hours (Including Exam Week: 16 x Total Hours)
Laboratory
Application
Special Course Internship
Field Work
Study Hours Out of Class 14 3 42
Presentation/Seminar Prepration
Project
Report
Homework Assignments 5 3 15
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury 1 10 10
Prepration of Final Exams/Final Jury 1 10 10