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Ordinary Differential Equation
Introduction and Preliminaries
There are many branches of science and engineering where differential equations arise naturally.
Now days, it finds applications in many areas including medicine, economics and social
sciences. In this context, the study of differential equations assumes importance. In addition, in
the study of differential equations, we also see the applications of many tools from analysis and
linear algebra. Without spending more time on motivation, (which will be clear as we go along)
let us start with the following notations. Let
be an independent variable and let
dependent variable of
(with respect to
. The derivatives of
The independent variable will be defined for an interval
be a
) are denoted by
where
is either
or an interval
with these notations, we are ready to define the term ``differential equation".
A differential equation is a relationship between the independent variable and the unknown
dependent function along with its derivatives. More precisely, we have the following definition.
DEFINITION 7.1.1 (Ordinary Differential Equation, ODE) An equation of the form
(7.1.1)
is called an ORDINARY DIFFERENTIAL EQUATION; where
is a known function from
to
Remark 7.1.2
1. The aim of studying the ODE (7.1.1) is to determine the unknown function
satisfies the differential equation under suitable conditions.
2. Usually (7.1.1) is written as
in most of the examples.
and the interval
which
is not mentioned
Some examples of differential equations are
1

1.
2.
3.
4.
5.
6.
7.
8.
DEFINITION 7.1.3 (Order of a Differential Equation) The ORDER of a differential equation
is the order of the highest derivative occurring in the equation.
In Example 7.1, the order of Equations 1, 3, 4, 5 are one, that of Equations 2, 6 and 8 are two and
the Equation 7 has order three.
DEFINITION 7.1.4 (Solution) A function
equation (7.1.1) on if
is called a SOLUTION of the differential
1.
is differentiable (as many times as the order of the equation) on
2.
satisfies the differential equation for all
for all
If
is a solution of an ODE (7.1.1) on
Sometimes a solution
and
. That is,
.
, we also say that
satisfies (7.1.1).
is also called an INTEGRAL.
EXAMPLE 7.1.5
1. Consider the differential equation
, then
is differentiable,
on
. We see that if we take
and therefore
2

Hence,
is a solution of the given differential equation for all
2. It can be easily verified that for any constant
differential equation
on any interval that does not contain the point
is a solution of the
as the function
defined at
. Furthere it can be shown that
equation whenever the interval contains the point
3. Consider the differential equation
that a solution
.
is not
is the only solution for this
.
on
. It can be easily verified
of this differential equation satisfies the relation
.
DEFINITION 7.1.6 (Explicit/Implicit Solution) A solution of the form
EXPLICIT SOLUTION (e.g., see Examples 7.1.5.1 and 7.1.5.2). If
is called an
is given by an implicit
relation
and satisfies the differential equation, then is called an IMPLICIT
SOLUTION (e.g., see Example 7.1.5.3).
Remark 7.1.7 Since the solution is obtained by integration, we may expect a constant of
integration (for each integration) to appear in a solution of a differential equation. If the order
of the ODE is
we expect
arbitrary constants.
To start with, let us try to understand the structure of a first order differential equation of the
form
(7.1.2)
and move to higher orders later.
3

DEFINITION 7.1.8 (General Solution) A function
is called a general solution of
(7.1.2) on an interval
if
is a solution of (7.1.2) for each
constant .
Remark 7.1.9 The family of functions
for an arbitrary
is called a one parameter family of functions and is called a parameter. In other words, a
general solution of (7.1.2) is nothing but a one parameter family of solutions of (7.1.2).
EXAMPLE 7.1.10
1. Determine a differential equation for which a family of circles with center at
arbitrary radius, is an implicit solution.
Solution: This family is represented by the implicit relation
and
(7.1.3)
2.
3. where
is a real constant. Then
is a solution of the differential equation
(7.1.4)
4.
5. The function satisfying (7.1.3) is a one parameter family of solutions or a general
solution of (7.1.4).
6. Consider the one parameter family of circles with center at
family is represented by the implicit relation
and unit radius. The
(7.1.5)
7.
8. where is a real constant. Show that satisfies
Solution: We note that, differentiation of the given equation, leads to
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