Course info

The module provides an overview of standard methods for solving single ordinary differential equations and systems of ordinary differential equations, with an introduction to the underlying theory. The first part is devoted to basic theory and methods for solving scalar ODEs. The second part of the module is devoted to the study of Systems of linear ODEs.

At the end of this module students should be able to:

1. use some of the standard methods to solve first order ODEs;
2. use some of the standard methods to solve second order ODEs;
3. be familiar with the basic theory and be able to solve higher order linear ODEs;
4. be familiar with the basic theory and be able to solve systems of first order linear ODEs.
5. be aware of the implications of existence and uniqueness theorems.

Syllabus:

1. Introduction, Classification of Differential Equations, First order differential equations: Linear Equations with Variable Coefficients, Separable Equations.

2. First order differential equations: Differences Between Linear and Nonlinear Equations, Exact Equations and Integrating Factors, (Euler) homogeneous equations, The Existence and Uniqueness Theorem.

3. Second Order Linear Equations: Homogeneous Equations with Constant Coefficients, Fundamental Solutions of Linear Homogeneous Equations, Linear Independence and the Wronskian, Complex Roots of the Characteristic Equation, Repeated Roots; Reduction of Order;The Existence and Uniqueness Theorem.

4. Second Order Linear Equations: ODEs with missing variables, Non-homogeneous Equations, Method of Undetermined Coefficients, Variation of Parameters.

5. Higher Order Linear Equations: General Theory of nth Order Linear Equations, Homogeneous Equations with Constant Coefficients, The Method of Undetermined Coefficients, The Method of Variation of Parameters.

6. Systems of First Order Linear Equations: Basic Theory of Systems of First Order Linear Equations including Linear Systems with Constant Coefficients.