1.
In this document you can review important
• Definitions
• Facts
• Procedures
Class –XII, CBSE MB1209: Differential Equations
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2.
What is Differential
Equation?
Class – XII,CBSE MB1209: Differential Equations
Important Definitions
Topic: Differential Equations
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3.
What is Differential
Equation?
Class – XII,CBSE
Important Definitions
An equation involving derivatives of
the dependent variable with respect to
independent variable (variables) is called
a .differential equation
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Topic: Differential EquationsMB1209: Differential Equations
4.
What is ordinary
Differential
Equation?
Class –XII,CBSE
Important Definitions
Learn Math for Free. Visit www.ganitgurooz.com
Topic: Differential EquationsMB1209: Differential Equations
5.
What is ordinary
Differential
Equation?
Class – XII,CBSE
Important Definitions
A differential equation involving
derivative of the dependent variable
with respect to only one independent
variable is called an
.
ordinary differential
equation
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Topic: Differential EquationsMB1209: Differential Equations
6.
What is the order
differential
equation?
Class – XII,CBSE
Important Definitions
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Topic: Differential EquationsMB1209: Differential Equations
7.
What is the order
differential
equation?
Class – XII,CBSE
Important Definitions
Topic: Differential Equations
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of a differential equation is the
order of the highest order derivative
occurring in the differential equation.
Order
MB1209: Differential Equations
8.
What is the degree
differential
equation?
Class – XII,CBSE
Important Definitions
Topic: Differential Equations
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MB1209: Differential Equations
9.
What is the degree
differential
equation?
Class – XII,CBSE
Important Definitions
Topic: Differential Equations
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of a differential equation is the
highest power (positive integer only)
of the highest order derivative in it.
Degree
MB1209: Differential Equations
10.
What is the solution
of differential
equation and its
types?
Class – XII,CBSE
Important Definitions
Topic: Differential Equations
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MB1209: Differential Equations
11.
What is the solution
of differential
equation and its
types?
Class – XII,CBSE
Important Definitions
Topic: Differential Equations
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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
MB1209: Differential Equations
12.
What is a
homogeneous
function?
Class – XII,CBSE
Important Definitions
Topic: Differential Equations
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MB1209: Differential Equations
13.
What is a
homogeneous
function?
Class – XII,CBSE
Important Definitions
Topic: Differential Equations
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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
MB1209: Differential Equations
14.
What is a
homogeneous
differential
equation?
Class – XII,CBSE
Important Definitions
Topic: Differential Equations
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MB1209: Differential Equations
15.
What is a
homogeneous
differential
equation?
Class – XII,CBSE
Important Definitions
Topic: Differential Equations
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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
MB1209: Differential Equations
16.
What is a first order
linear differential
equation?
Class – XII,CBSE
Important Definitions
Topic: Differential Equations
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MB1209: Differential Equations
17.
What is a first order
linear differential
equation?
Class – XII,CBSE
Important Definitions
Topic: Differential Equations
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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
MB1209: Differential Equations
18.
Fact # 1
Class – XII,CBSE
Important Facts
Topic: Differential Equations
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of a differential equation is defined if it is a polynomial equation in its
derivatives.
Degree
MB1209: Differential Equations
19.
Fact # 2
Class – XII,CBSE
Important Facts
Topic: Differential Equations
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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.
MB1209: Differential Equations
20.
Fact # 3
Class – XII,CBSE
Important Facts
Topic: Differential Equations
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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
MB1209: Differential Equations
21.
How to form a differential equation?
Class – XII,CBSE
Important Procedures
Topic: Differential Equations
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MB1209: Differential Equations
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
22.
How to form a differential equation?
Class – XII,CBSE
Important Procedures
Topic: Differential Equations
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MB1209: Differential Equations
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
23.
How to form a differential equation?
Class – XII,CBSE
Important Procedures
Topic: Differential Equations
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MB1209: Differential Equations
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
24.
How to solve first order, first degree differential equation by
variable separable method?
Class – XII,CBSE
Important Procedures
Topic: Differential Equations
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MB1209: Differential Equations
25.
How to solve first order, first degree differential equation by
variable separable method?
Class – XII,CBSE
Important Procedures
Topic: Differential Equations
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MB1209: Differential Equations
: 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
26.
How to solve first order, first degree differential equation by
variable separable method?
Class – XII,CBSE
Important Procedures
Topic: Differential Equations
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MB1209: Differential Equations
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
27.
How to solve first order, first degree differential equation by
variable separable method?
Class – XII,CBSE
Important Procedures
Topic: Differential Equations
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MB1209: Differential Equations
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
28.
How to solve first order, first degree homogeneous
differential equation?
Class – XII,CBSE
Important Procedures
Topic: Differential Equations
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MB1209: Differential Equations
: The differential equation is of the type
F( , ) ...... 1
Substitute ...... 2
dy y
x y g
dx x
y v x
Step1
29.
How to solve first order, first degree homogeneous
differential equation?
Class – XII,CBSE
Important Procedures
Topic:
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MB1209: Differential Equations
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
30.
How to solve first order, first degree homogeneous
differential equation?
Class – XII,CBSE
Important Procedures
Topic: Differential Equations
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MB1209: Differential Equations
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
31.
How to solve first order, first degree homogeneous
differential equation?
Class – XII,CBSE
Important Procedures
Topic: Differential Equations
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MB1209: Differential Equations
The general solution of the differential equation (1) is given by equation (6)
when we replace by .
y
v
x
32.
How to solve first order, first degree homogeneous
differential equation?
Class – XII,CBSE
Important Procedures
Topic: Differential Equations
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MB1209: Differential Equations
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
33.
How to solve first order linear differential equation?
Class – XII,CBSE
Important Procedures
Topic: Differential Equations
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MB1209: Differential Equations
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
34.
How to solve first order, first degree linear differential
equation?
Class – XII,CBSE
Important Procedures
Topic: Differential Equations
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MB1209: Differential Equations
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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