He who seeks for methods without having a definite problem in mind
seeks for the most part in vain. – D. HILBERT
9.1
Introduction
In Class XI and in Chapter 5 of the present book, we
discussed how to differentiate a given function
f
with respect
to an independent variable, i.e., how to find
f
′
(
x
) for a given
function
f
at each
x
in its domain of definition. Further, in
the chapter on Integral Calculus, we discussed
how to find
a function
f
whose derivative is the function
g
, which may
also be formulated as follows:
For a given function
g
, find a function
f
such that
dy
dx
=
g
(
x
), where
y = f
(
x
)
... (1)
An equation of the form (1) is known as a
differential
equation
. A formal definition will be given later.
These equations arise in a variety of applications, may it be in Physics, Chemistry,
Biology, Anthropology, Geology,
Economics etc. Hence, an indepth study of differential
equations has assumed prime importance in all modern scientific investigations.
In this chapter, we will study some basic concepts related to differential equation,
general and particular solutions of a differential equation, formation of differential
equations, some methods to solve a first order  first degree differential equation and
some applications of differential equations in different areas.
9.2
Basic Concepts
We are already familiar with the equations of the type:
x
2
– 3
x
+ 3 = 0
... (1)
sin
x
+ cos
x
= 0
... (2)
x
+
y
= 7
... (3)
Chapter
9
DIFFERENTIAL
EQUATIONS
Henri Poincare
(18541912 )
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MATHEMATICS
380
Let us consider the equation:
dy
x
y
dx
±
= 0
... (4)
We see that equations (1), (2) and (3) involve independent and/or dependent variable
(variables) only but equation (4) involves variables as well as derivative of the dependent
variable
y
with respect to the independent variable
x
. Such an equation is called a
differential equation
.
In general, an equation involving
derivative (derivatives) of the dependent variable
with respect to independent variable (variables) is called a differential equation.
A differential equation involving derivatives of the dependent variable with respect
to only one independent variable is called an ordinary differential equation, e.g.,
3
2
2
2
d y
dy
dx
dx
§
·
±
¨
¸
©
¹
= 0
is an ordinary differential equation
....
(5)
Of course, there are differential equations involving derivatives with respect to
more than one independent variables, called partial differential equations but at this
stage we shall confine ourselves to the study of ordinary differential equations only.
Now onward, we will use the term ‘differential equation’ for ‘ordinary differential
equation’.
$
Note
1.
We shall prefer to use the following notations for derivatives:
2
3
2
3
,
,
dy
d y
d y
y
y
y
dx
dx
dx
c
cc
ccc
2.
For derivatives of higher order, it will be inconvenient
to use so many dashes
as supersuffix therefore, we use the notation
y
n
for
n
th order derivative
n
n
d y
dx
.
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 Winter '08
 Hrebian
 dx, L.H.S.

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