MATH587 ORDINARY DIFFERENTIAL EQUATIONS I
| Course Code: | 2360587 |
| METU Credit (Theoretical-Laboratory hours/week): | 3 (3.00 - 0.00) |
| ECTS Credit: | 8.0 |
| Department: | Mathematics |
| Language of Instruction: | English |
| Level of Study: | Graduate |
| Course Coordinator: | Prof.Dr. MARAT AKHMET |
| Offered Semester: | Fall Semesters. |
Course Objectives
Ordinary differential equations are crucial in the fields of science and engineering, as they provide mathematical models for many real-world phenomena. This significance underscores the importance of studying differential equations. Consequently, it is essential to teach both the theory and applications of these equations to students pursuing advanced studies in applied sciences. This course aims to systematically introduce the qualitative theory of differential equations at an introductory level. The course content will familiarize students with the fundamental principles and methods of modern ordinary differential equations.
Course Content
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; Linearization; Stability of an Equilibrium and Stable Manifold for Nonautonomous Differential Equations.
Course Learning Outcomes
By the end of the course, the student will learn the following components of the theory:
- Domains for the differential equations, integral curves, trajectories, and phase portraits.
- The definitions of local and global existence and uniqueness of solutions. Examples of the phenomena in models.
- Theorems of existence and uniqueness of solutions with various sufficient conditions. The instance of nowhere uniqueness.
- Definitions of continuous and differentiable dependence of solutions on initial data and parameters.
- Theorems on the dependence of solutions on initial data and parameters.
- Method of the small parameter in the non-critical case.
- Analysis of linear homogeneous and inhomogeneous systems of differential equations, focusing on the methods of solutions, description of individual and collective behaviour. Spaces of solutions.
- Periodic systems of linear equations, criteria of existence for periodic solutions.
- Higher-order linear equations. Connection with the first-order linear systems. The initial value problem.
- Definitions of Lyapunov stability. Linear, quasilinear systems. Stability by linearization.
- Lyapunov’s second method. Application to mechanical and electrical models.
Program Outcomes Matrix
| Level of Contribution | |||||
| # | Program Outcomes | 0 | 1 | 2 | 3 |
| 1 | Acquires mathematical thinking skills (problem solving, generating ways of thinking, forming correspondence, generalizing etc.) and can use them in related fields. | ✔ | |||
| 2 | Gains academic maturity through self-study. | ✔ | |||
| 3 | Can design mathematics related problems, devise solution methods and apply them when appropriate. | ✔ | |||
| 4 | Carries out parts of a mathematical research program independently. | ✔ | |||
| 5 | Has a command of Turkish and English languages so that he/she can actively communicate (read, write, listen and speak). | ✔ | |||
| 6 | Contributes to solving global, environmental and social problems either individually or as being part of a social group. | ✔ | |||
| 7 | Respects ethical values and rules; applies them in professional and social issues. | ✔ | |||
| 8 | Can work cooperatively in a team and also individually. | ✔ | |||
| 9 | Gets exposed to academic culture through interaction with others. | ✔ | |||
| 10 | Comprehends necessity of knowledge, can define it and acquires it; uses knowledge effectively and shares it with others. | ✔ | |||
0: No Contribution 1: Little Contribution 2: Partial Contribution 3: Full Contribution