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Class –XII, CBSE MB1209: Differential Equations
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What is Differential
Equation?
Class – XII,CBSE MB1209: Differential Equations
Important Definitions
Topic: Differential Equations

What is Differential
Equation?
Class – XII,CBSE
An equation involving derivatives of
the dependent variable with respect to
independent variable (variables) is called
a .differential equation
Topic: Differential EquationsMB1209: Differential Equations

What is ordinary
Differential
Equation?
Class –XII,CBSE

What is ordinary
Differential
Equation?
Class – XII,CBSE
A differential equation involving
derivative of the dependent variable
with respect to only one independent
variable is called an
.
ordinary differential
equation

What is the order
differential
equation?
Class – XII,CBSE

What is the order
differential
equation?
Class – XII,CBSE
of a differential equation is the
order of the highest order derivative
occurring in the differential equation.
Order
MB1209: Differential Equations

What is the degree
differential
equation?
Class – XII,CBSE

What is the degree
differential
equation?
Class – XII,CBSE
of a differential equation is the
highest power (positive integer only)
of the highest order derivative in it.
Degree

What is the solution
of differential
equation and its
types?
Class – XII,CBSE

What is the solution
of differential
equation and its
types?
Class – XII,CBSE
A function which satisfies the given
differential equation is called its .
The solution which contains as many
arbitrary constants as the order of the
differential equation is called a
solution
general
and the solution free from
arbitrary constants is called
.
solution
particular
solution

What is a
homogeneous
function?
Class – XII,CBSE

What is a
homogeneous
function?
Class – XII,CBSE
A function ( , ) is a homogeneous
function of deg ree if,
F( , ) orn n
F x y
n
y x
x y x g y h
x y

What is a
homogeneous
differential
equation?
Class – XII,CBSE

What is a
homogeneous
differential
equation?
Class – XII,CBSE
A differential equation which can be
expressed in the form ( , )
or ( , ) where, ( , ) and
( , ) are homogenous functions of
degree zero is called a
.
dy
f x y
dx
dx
g x y f x y
dy
g x y
homogeneous
differential equation

What is a first order
linear differential
equation?
Class – XII,CBSE

What is a first order
linear differential
equation?
Class – XII,CBSE
A differential equation of the form
P Q,
where P and Q are constants or
functions of only is called a first
order .
dy
x
dx
x
linear differential equation

Fact # 1
Class – XII,CBSE
Important Facts
of a differential equation is defined if it is a polynomial equation in its
derivatives.
Degree

Fact # 2
Class – XII,CBSE
Important Facts
To form a differential equation from a given function we differentiate the function
successively as many times as the number of arbitrary constants in the given function
and then eliminate the arbitrary constants.

Fact # 3
Class – XII,CBSE
Important Facts
Variable separable method is used to solve such an equation in which variables can
be separated completely i.e. terms containing should remain with and terms
containing should remain with .
y dy
x dx

How to form a differential equation?
Class – XII,CBSE
Important Procedures
1
1
:
If the given family F of curves depend on only one parameter then it is
represented by the equation of the form ( , , ) 0.
Differentiate the above equation with respect to x a
One parameter
F x y a
Step1
nd get an equation of
the form ( , , , ) 0.
The required differential equation is obtained by eliminating a from the
equation as ( , , ) 0.
g x y y a
F x y y

Class – XII,CBSE
2
2
:
If the given family F of curves depend on the parameters , (say) then
it is represented by the equation of the form F ( , , , ) 0 ......(1)
Differentiate equation (1) with resp
Two parameter
a b
x y a b
Step 2
ect to x, we get an equation of the form
( , , , , ) 0 ......(2)
Differential equation (2) with respect to x to get an equation of the form
( , , , , , ) 0 ......(3)
g x y y a b
h x y y y a b

Class – XII,CBSE
The required differential equation is obtained by eliminating and from
the equation (1),(2) and (3) as ( , , , ) 0.
a b
F x y y y

How to solve first order, first degree differential equation by
variable separable method?
Class – XII,CBSE

Class – XII,CBSE
: A first order first degree differential equation is of the form,
F( , ) ...... 1
Express ( , ) as a product ( ) ( ), where ( ) is a function of
and ( ) is a function of . The different
dy
x y
dx
F x y g x h y g x x
h y y
Step 1
ial equation (1) then has the
form: ( ) ( ) ...... 2
dy
h y g x
dx

Class – XII,CBSE
If ( ) 0, separating the variables, (2) can be rewritten as
1
( ) ...... 3
( )
Integrate both sides of (3), we get
1
( ) ...... 4
( )
h y
dy g x dx
h y
dy g x dx
h y

Class – XII,CBSE
Thus, the solution of the given differential equation is of the form
( ) ( ) , where ( ) and ( ) are the anti derivatives if
1
and ( ) respectively and is the arbitrary constant.
( )
H y G x C H y G x
g x C
h y

How to solve first order, first degree homogeneous
differential equation?
Class – XII,CBSE
: The differential equation is of the type
F( , ) ...... 1
Substitute ...... 2
dy y
x y g
dx x
y v x
Step1

Class – XII,CBSE
Topic:
Differentiate equation (2) with respect to , we get
...... 3
Substitute the value of from equation (3) in equation (1), and get
( ) ...... 4
x
dy dv
v x
dx dx
dy
dx
dv
x g v v
dx

Class – XII,CBSE
Separating the variables in equation (4), we get
...... 5
( )
Integrating both sides of equation (5) we get
...... 6
( )
dv dx
g v v x
dv dx
C
g v v x

Class – XII,CBSE
The general solution of the differential equation (1) is given by equation (6)
when we replace by .
y
v
x

Class – XII,CBSE
If the homogeneous differential is of the form F( , ), where ( , )
is a homogeneous function of degree zero , then we make the substitution
and proceed further to find the general solutio
dx
x y F x y
dy
x v y n as discussed above
by writing F( , ) h .
dx x
x y
dy y

How to solve first order linear differential equation?
Class – XII,CBSE
P
: The given differential equation in the form
P Q where P and Q are constants or functions of only.
Find the integrating factor (I.F) = e .
Write the solution of the given differenti
dx
dy
y x
dx
Step1
al equation as
I.F (Q I.F) .y dx C

How to solve first order, first degree linear differential
equation?
Class – XII,CBSE
1
1 1 1 1
P
: The given differential equation is of the form
P Q where P and Q are constants or functions of only.
Find the integrating factor (I.F) = e .
Write the solution of the given diff
dy
dx
x y
dy
Step 2
1
erential equation as
. I.F (Q I.F) .x dy C

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