# 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

Requirements Number Percentage of Grade
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 Accumulated knowledge on mathematics, science and mechatronics engineering; an ability to apply the theoretical and applied knowledge of mathematics, science and mechatronics engineering to model and analyze mechatronics engineering problems. X
2 An ability to differentiate, identify, formulate, and solve complex engineering problems; an ability to select and implement proper analysis, modeling and implementation techniques for the identified engineering problems. X
3 An ability to design a complex system, product, component or process to meet the requirements under realistic constraints and conditions; an ability to apply contemporary design methodologies; an ability to implement effective engineering creativity techniques in mechatronics engineering. (Realistic constraints and conditions may include economics, environment, sustainability, producibility, ethics, human health, social and political problems.) X
4 An ability to develop, select and use modern techniques, skills and tools for application of mechatronics engineering and robot technologies; an ability to use information and communications technologies effectively.
5 An ability to design experiments, perform experiments, collect and analyze data and assess the results for investigated problems on mechatronics engineering and robot technologies.
6 An ability to work effectively on single disciplinary and multi-disciplinary teams; an ability for individual work; ability to communicate and collaborate/cooperate effectively with other disciplines and scientific/engineering domains or working areas, ability to work with other disciplines.
7 An ability to express creative and original concepts and ideas effectively in Turkish and English language, oral and written.
8 An ability to reach information on different subjects required by the wide spectrum of applications of mechatronics engineering, criticize, assess and improve the knowledge-base; consciousness on the necessity of improvement and sustainability as a result of life-long learning; monitoring the developments on science and technology; awareness on entrepreneurship, innovative and sustainable development and ability for continuous renovation.
9 Be conscious on professional and ethical responsibility, competency on improving professional consciousness and contributing to the improvement of profession itself.
10 A knowledge on the applications at business life such as project management, risk management and change management and competency on planning, managing and leadership activities on the development of capabilities of workers who are under his/her responsibility working around a project.
11 Knowledge about the global, societal and individual effects of mechatronics engineering applications on the human health, environment and security and cultural values and problems of the era; consciousness on these issues; awareness of legal results of engineering solutions.
12 Competency on defining, analyzing and surveying databases and other sources, proposing solutions based on research work and scientific results and communicate and publish numerical and conceptual solutions.
13 Consciousness on the environment and social responsibility, competencies on observation, improvement and modify and implementation of projects for the society and social relations and be an individual within the society in such a way that planing, improving or changing the norms with a criticism.
14 A competency on developing strategy, policy and application plans on the mechatronics engineering and evaluating the results in the context of qualitative processes.

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