# Differential Equations I (MAE614)

### General

School School of Science Department of Mathematics Undergraduate MAE614 6 Differential Equations I Lectures (Weekly Teaching Hours: 3, Credits: 6) Special Background - Language of Instruction (lectures): Greek Language of Instruction (activities other than lectures): Greek and English Language of Examinations: Greek and English Yes See eCourse, the Learning Management System maintained by the University of Ioannina.

### Learning Outcomes

Learning outcomes Remembering: The notion of a system of first order differential equations. The notion of a linear system of differential equations. The notion of a homogenous linear system of differential equations. The notion of a fundamental set of solutions for a system of differential equations. The notion of the fundamental matrix of a system of differential equations. The characteristic polynomial for a system of differential equations with constant coefficients. Similarity of matrices. Jordan canonical form. The notion of autonomous systems. The notion of the phase space. The notion of the critical point for a system of differential equations. The notion of the proper node, the improper node, the saddle point and the spiral point for the phase space. The notion of stability for systems of differential equations. The notion of the equilibrium point. The notions of the asymptotic stability and the instability of equilibrium points. The notion of asymptotic equivalence of systems of differential equations. The notion of the dynamical system. Comprehension: Conditions for the existence of solutions for a differential equation. Conditions for the uniqueness of the solution for a differential equation. Finding the maximum domain of the solutions for a differential equation. Conditions for the existence of solutions for a system of differential equations. Conditions for the uniqueness of the solution for a system of differential equations. Finding the maximum domain of the solutions for a system of differential equations. The Gronwall inequality. The method known as “variation of constants” for solving systems of differential equations. Study of the asymptotic behaviour of solutions of systems of differential equations. Study of autonomous systems. Finding the phase space for solutions of systems of differential equations. Study of linear systems of differential equations with periodic coefficients. Dependence of the solutions of systems of differential equations on the initial conditions. Study of the stability for systems of differential equations. Using the Lyapunov method in order to study the stability of systems of differential equations. Studying dynamical systems. Applying: Studying oscillations which do not tend toward a state of rest. Studying the motion of a pendulum. Studying predator - prey models. Evaluating: Teaching undergraduate and graduate courses. Creative, analytical and inductive thinking. Required for the creation of new scientific ideas. Working independently. Working in groups. Decision making.

### Syllabus

This course is designed to be the continuation of the compulsory course “Introduction to Differential Equations” and is comprised of two main, closely related parts. The first part consists of the study of ordinary differential equations regarding their qualitative properties. The second part studies the methods of solving as well as the study of the qualitative properties of systems of ordinary differential equations.
In the first part, initial value problems for ordinary differential equations are studied regarding the existence, uniqueness, continuation and dependence on the initial conditions of their solutions.
The second part consists of the study of systems of ordinary differential equations. Specifically, this part studies the existence and the uniqueness of solutions, algorithms for finding solutions of specific forms of such systems, the phase space and the stability of solutions. Also, the well-known Lyapunov Theorems are presented.
For each part, specific real-life applications are presented.

### Teaching and Learning Methods - Evaluation

Delivery
• Lectures in class.
• Teaching is assisted by Learning Management System.
• Teaching is assisted by the use of online forums where students can participate in order to improve their problem solving skills, as well as their understanding of the theory they are taught.
• Teaching is assisted by the use of pre-recorded videos.
Use of Information and Communications Technology
• Use of Learning Management System, combined with File Sharing Platform as well as Blog Management System for
1. distributing teaching material,
2. submission of assignments,
3. course announcements,
4. gradebook keeping for all students evaluation procedures,
5. communicating with students.
• Use of Appointment Scheduling System for organising appointments between students and the teacher.
• Use of Survey Web Application for submitting anonymous evaluations regarding the teacher.
• Use of Wiki Engine for publishing manuals regarding the regulations applied at the exams processes, the way teaching is organized, the grading methods, as well as the use of the online tools used within the course.
Teaching Methods
Lectures (7x3) 21
Seminars (6x3) 18
Individual study 78
Exrecises/projects 33
Course total 150
Student Performance Evaluation

Language of evaluation: Greek and English. Methods of evaluation:

• Weekly written assignments.
• Few number of tests during the semester.
• Based on their grades in the aforementioned weekly assignments and tests, limited number of students can participate in exams towards the end of the semester, before the beginning of the exams period.

In any case, all students can participate in written exams at the end of the semester, during the exams period. The aforementioned information along with all the required details are available through the course's website. The information is explained in detail at the beginning of the semester, as well as, throughout the semester, during the lectures. Reminders are also posted at the beginning of the semester and throughout the semester, through the course's website. Upon request, all the information is provided using email or social networks.

### Attached Bibliography

See the official Eudoxus site or the local repository of Eudoxus lists per academic year, which is maintained by the Department of Mathematics. Books and other resources, not provided by Eudoxus:

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