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DIFFERENTIAL EQUATION/(D.E.)
(I) Differential equation: A differential equation is an equation which
contains derivatives of various order of dependent variable with respect
to independent variables.
A general differential equation is of the type:
2
3
n
 d

y
d
y
d
yd
y
fx
,,
y
,
,
,
.
.
.
,

0


2
3
n
x
d
xd
x d
x
 d

NOTE: Alternative symbols for various order derivatives
2
n
d
y /
y //
d
y n
/ d
/
/
n

f 
x
y

y

f
x

y

y

f 
x
y

y

;
;…;
1
2
n
2
n
d
x
d
x
d
x
Examples of differential equation
2
d
y
d
y
5
x
y 
6
y9

0
1. x 2
d
x
d
x
2
1

xy
x
y
y

4
50

3
2. 
2
2
(II) Order and degree of the differential equation
Order of the differential equation: Order of the differential equation is
the order of the highest order of the derivatives appearing in the
differential equation.
Degree of the differential equation: Degree of the differential equation is
the highest power of the highest order derivative appearing in the
differential equation
NOTE:
1. Degree of the differential equation is to be found only after removing
all negative, fractional or radical powers from the differential equation.
2. Degree of a differential equation can only be defined if it can be
expressed as the polynomial of derivatives.
2
2
dy

y
d
e.g. Degree of the differential equation  2 sin can’t be defined
x
x
d
d
 
as it contains the term sin  dx  therefore it is not a polynomial of
 
derivatives, although it has order ‘2’.
dy
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(III) FORMATION OF A DIFFERENTIAL EQUATION
In order to form a differential equation from a given equation
containing ‘n’ arbitrary constants we proceed as follows
STEP 1: Differentiate the given differential equation as many times as it
contains the arbitrary constants.
STEP 2: From the ‘n+1’ differential equation (obtained by
differentiating the given equation ‘n’ times and one the equation as
follows) obtained from step1, we make an equation free from all
arbitrary constant
The equation obtained in step 2 is the required differential equation.
(IV) SOLVING A DIFFERENTIAL EQUATION
TYPE (1). VARIABLE SEPERABLE:
A differential equation of the type or which can be expressed as
d
y
fx.gy

d
x
is called as variable separable.
METHOD OF SOLVING
  yfxd
x
STEP 1: Express the D.E. as gyd
STEP 2: Integrate both sides of the equation obtained in step 1.
y
xd
x
y
d
i.e. find g
f
The solution obtained in step 2 is the required solution of the given
differential equation.
TYPE (2). HOMOGENEOUS DIFFERENTIAL EQUATION
A differential equation which is not a variable separable, say,
dy
 f x, y
dx
will be called as homogeneous differential equation iff we can express
n
f

x
,y

f
x
,y

, for some n  W , in case there exists a ‘n’, we say that
the D.E. is a homogeneous D.E. of degree ‘n’.
EXAMPLES:
d
y

x
2
y.
1. xyd
x
2
2
d
y

y
d
x
x

y
d
xetc. are the homogeneous
2. x
differential equations.
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NOTE: A homogeneous differential equation can also be written as
d
y
y 
x


fx

g
o
r
h
,y



d
x
x 
y

i.e. we can express the algebraic expression f  x, y  in the differential
y
x
equation as an expression of x or y .
EXAMPLE 2.
The differential equations in example 1 can be written as
dy
 x 2y
dx
dy x  2 y


dx
x y
x
x
 2
 2
dy
y y
y



x
dx
y x
1
1
y
y
a lte r n a te ly ,
2y
2y
1
1
dy x
x 
x

y
y
dx x
1
1
x
x
x
 y
METHOD OF SOLVING A HOMOGENEOUS DIFFERENTIAL
EQUATION
STEP 1. Put y  vx or x  vy (as the situation may be)
dy
dv
dx
dv
STEP 2. Put dxvxdxor dyvydy.
STEP 3. After step 3 we simplify the equation and bring it to variable
separable form and hence solve it as discussed in variable separable
form.
y
x
Finally, we replace v  x or v  y as the case may be.
FOR THE SOLUTION PLEASE TURN OVER
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TYPE (3). LINEAR ORDER DIFFERENTIAL EQUATION
A linear order differential equation is of two type
Type 1. A differential equation which can be written as
dy
PyQ……………….(1)
dx
Where, P and Q are either constants or functions of ‘x’ only
is a linear order differential equation.
SOLUTION
The solution to linear order D.E.(1) is given by
y

I
..
F


I
..
F
d
x

Q

Where, I
..
F
(
I
n
t
e
g
r
a
t
i
n
g
f
a
c
t
o
r
)

e
Type 2. A differential equation which can be written as
P
d
x
dx
PxQ ……………….(2)
dy
Where, P and Q are either constants or functions of ‘y’ only
is a linear order differential equation.
SOLUTION
The solution to linear order D.E.(2) is given by
x

I
..
F


I
..
F
d
y

Q

Where, I
..
F
(
I
n
t
e
g
r
a
t
i
n
g
f
a
c
t
o
r
)

e
P
d
y
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Prepared and designed by Ajay Marwaha