Ordinary Differential Equation

Ordinary Differential Equations
[FDM 1023]

Chapter 1

Introduction to Ordinary
Differential Equations

Chapter 1:

Introduction to Differential Equations

Overview
1.1. Definitions
1.2. Classification of Solutions
1.3. Initial and Boundary Value Problems

1.1. Definitions

Learning Outcomes
At the end of the section, you should be able to:
1) Define a differential equation
2) Classify differential equations by type, order and linearity

Recall
Dependent and Independent variables?

Example 1
Consider an equation y = 4 x + 1
The values of y depend on the choice of x. y - dependent variable (function) x - independent variable (argument)

Example 2
For a parametric equations of the unit circle

The values of y and x depend on the choice of θ. y and x - dependent variable θ - independent variable

What is

represents?

It represents the derivative of dependent variable y with respect to independent variable x.
By first principle, it is defined as rate of change of y with respect to x.

1.1. Definitions

What is a Differential Equation?
A differential equation (DE) is an equation containing the derivatives of one or more dependent variables with respect to one or more independent variables.

1.1. Definitions

Examples

1.1. Definitions

Applications
1) Population growth
2) Radioactive decay
3) Chemical reactions
4) Spread of a disease
5) Series circuits

1.1. Definitions

1.1. Definitions

Classification
DE can be classified by:
• TYPE
• ORDER
• LINEARITY

1.1. Definitions

Classification by Type
Two types of DE i.e.

1) Ordinary Differential Equations (ODE)
2) Partial Differential Equations (PDE)

1.1. Definitions

Ordinary Differential Equations
An equation containing only ordinary derivatives of one or more dependent variables with respect to a SINGLE independent variable is said to be an Ordinary
Differential Equation (ODE).

Examples

1.1. Definitions