ORDINARY DIFFERENCIAL
EQUATIONS (ODEs)
Hamid Bouhioui
The BOOK
we’re using
Chap 1- First Order ODE's
Many physical laws and relations can be expressed
mathematically in the form of differential equations. Indeed,
many engineering problems appear as differential equations.
Ordinary differential equations (ODEs) are differential
equations that depend on a single variable. Differential
equations that depend on several variables are called Partial
Differential Equations (PDEs) –Beyond the scope of our
course.
Introduction:
We begin the study of ordinary differential equations (ODEs) by
deriving them from physical problems (modeling), solving them
by standard mathematical methods, and interpreting solutions and
their graphs in terms of a given problem.
The simplest ODEs to be discussed are ODEs of the first order
because they involve only the first derivative of the unknown
function and no higher derivatives. These unknown functions will
usually be denoted by y(x) or y(t) when the independent variable
denotes time t.
Basic Concepts –Modeling:
If we want to solve an engineering problem (usually of a physical
nature), we first have to formulate the problem as a mathematical
expression in terms of variables, functions, and equations. Such
an expression is known as a mathematical model of the given
problem. That is mathematical modeling or, briefly, modeling.
Many physical concepts, such as velocity and acceleration, are
derivatives. Hence a model is very often an equation containing
derivatives of an unknown function.
Such a model is called a differential equation.
Examples
An ordinary differential equation (ODE) is an equation that
contains one or several derivatives of an unknown function, which
we usually call y(x) or y(t). The equation may also contain “y”
itself, known functions of x (or t), and constants. Note: y’=dy/dx
For example, these equations are ODEs
And this equation is a Partial Differential equation PDE:
Because the function u(x,y)
depends on 2 variables, x and y)
Order of a differential equation:
An ODE is said to be of order n if the nth derivative of the
unknown function y is the highest derivative of y in the equation.
The concept of order gives a useful classification into ODEs of
first order, second order, and so on. Thus, (1) is of first order, (2)
of second order, and (3) of third order.
In this chapter we consider First Order ODEs
An ODE can be written in an:
Explicit form : y’=f(x,y) for example:
Or
Implicit form : F(x,y,y’) = 0 for example:
Example of Solution of an ODE:
Examples of ODEs show that a solution contains an arbitrary
constant c. Such a solution is called a general solution of the
ODE.
Geometrically, the general solution of an ODE is a family of
infinitely many solution curves, one for each value of the constant
c.
If we choose a specific c (e.g., c=1 or c=0 or …) we obtain what is
called a particular solution of the ODE. A particular solution
does not contain any arbitrary constants.
In most cases, general solutions exist, and every solution not
containing an arbitrary constant is obtained as a particular solution
by assigning a suitable value to c.
Initial Value Problem
In most cases the unique solution of a given problem, is obtained
from a general solution by an initial condition y(x0)=y0, with given
values x0 and y0, that is used to determine a value of the arbitrary
constant c.
An ODE, together with an initial condition, is called an initial
value problem. Thus, if the ODE is explicit, the initial value
problem is of the form:
y’=f(x,y) and y(x0)=y0
Example of an initial value problem:
