Ordinary Differential Equations

Prerequisite(s)

Fundamental Algebra and Analysis II

Course Description:

This course aims to provide an introduction to ordinary differential equations, including first order linear ordinary equations, higher order linear ordinary equations, systems of ordinary linear equations, and some famous ordinary equations in history. We will focus on two aspects of linear equations: how to solve ODEs implicitly or explicitly, and asymptotic analysis of solutions of ODEs.

This course requires background in elementary calculus (first-year undergraduate course in analysis) and linear algebra. Knowledge of basic topology in Euclidean space will be helpful but not required.

Learning Objectives:

By the end of this course, students will be able to:

1. Conceptual Understanding

Explain what an ordinary differential equation is and distinguish between order, linearity, and types of ODEs. Interpret differential equations as models of physical, biological, and engineering systems. Describe the meaning of initial value problems and boundary value problems.

2. Analytical Solution Techniques

Solve first-order ODEs using separation of variables, exact equations, and integrating factors.

Solve higher-order linear ODEs with constant coefficients. Solve nonhomogeneous linear ODEs using undetermined coefficients and variation of parameters. Solve systems of first-order linear ODEs using eigenvalues and eigenvectors.

3. Qualitative and Geometric Analysis

Analyze solution behavior using direction fields and phase portraits. Classify equilibria and determine stability and long-term behavior.

4. Modeling and Applications

Formulate ODE models for real-world problems. Interpret solutions in the context of applications and assess model assumptions. Apply basic numerical methods such as Euler's method.

5. Mathematical Communication

Present solutions clearly using correct notation, graphs, and explanations. Justify solution methods and interpret results coherentlygive them an introduction to mordern study of differential equations, to lay the foundations of further study in modern mathematics.

Detailed Content:

1 Phase spaces, integral curve of direction field, concept of differential equation and its solution

2 Example of ordinary differential equations: Some types of ODES, solvability and non solvability.

3 Uniqueness and nonuniquenes of general ODEs.

4 Theory of first order linear ODE: homogeneous case and theory of first order linear ODE: non homogeneous case.

5 Picard's Theorem: existence and uniqueness of solution of ODEs.

6 Comparison theorem and its application to theory of ODEs

7 An introduction to higher order ODEs and system of ODEs.

8 Review of basic knowledge of linear algebra

9 Explicit methods of solving system of linear ODEs with constant coefficient and explicit methods of solving system of linear ODEs

10 General theory of system of linear ODES

11 General theory of higher order ODEs:dependence of solution on the initial value, continuity and differentiability

12 Power series method and Cauchy theorem

13 Boundary value problem and Sturm comparison theorem.