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1 | 4418-4421 | (2)
x + y = 7 (3)
Chapter 9
DIFFERENTIAL EQUATIONS
Henri Poincare
(1854-1912 )
© NCERT
not to be republished
MATHEMATICS
380
Let us consider the equation:
xdy
dx +y
= 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 |
1 | 4419-4422 | (3)
Chapter 9
DIFFERENTIAL EQUATIONS
Henri Poincare
(1854-1912 )
© NCERT
not to be republished
MATHEMATICS
380
Let us consider the equation:
xdy
dx +y
= 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 |
1 | 4420-4423 | (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 |
1 | 4421-4424 | 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 |
1 | 4422-4425 | 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 d y2
dy
dx
dx
⎛
⎞
+ ⎜
⎟
⎝
⎠
= 0 is an ordinary differential equation |
1 | 4423-4426 | 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 d y2
dy
dx
dx
⎛
⎞
+ ⎜
⎟
⎝
⎠
= 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 |
1 | 4424-4427 | g ,
3
2
2 d y2
dy
dx
dx
⎛
⎞
+ ⎜
⎟
⎝
⎠
= 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’ |
1 | 4425-4428 | ,
3
2
2 d y2
dy
dx
dx
⎛
⎞
+ ⎜
⎟
⎝
⎠
= 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 |
1 | 4426-4429 | (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
′
′′
′′′
=
=
=
2 |
1 | 4427-4430 | 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
′
′′
′′′
=
=
=
2 For derivatives of higher order, it will be inconvenient to use so many dashes
as supersuffix therefore, we use the notation yn for nth order derivative
n
d yn
dx |
1 | 4428-4431 | �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
′
′′
′′′
=
=
=
2 For derivatives of higher order, it will be inconvenient to use so many dashes
as supersuffix therefore, we use the notation yn for nth order derivative
n
d yn
dx 9 |
1 | 4429-4432 | 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
′
′′
′′′
=
=
=
2 For derivatives of higher order, it will be inconvenient to use so many dashes
as supersuffix therefore, we use the notation yn for nth order derivative
n
d yn
dx 9 2 |
1 | 4430-4433 | For derivatives of higher order, it will be inconvenient to use so many dashes
as supersuffix therefore, we use the notation yn for nth order derivative
n
d yn
dx 9 2 1 |
1 | 4431-4434 | 9 2 1 Order of a differential equation
Order of a differential equation is defined as the order of the highest order derivative of
the dependent variable with respect to the independent variable involved in the given
differential equation |
1 | 4432-4435 | 2 1 Order of a differential equation
Order of a differential equation is defined as the order of the highest order derivative of
the dependent variable with respect to the independent variable involved in the given
differential equation Consider the following differential equations:
dy
dx = ex |
1 | 4433-4436 | 1 Order of a differential equation
Order of a differential equation is defined as the order of the highest order derivative of
the dependent variable with respect to the independent variable involved in the given
differential equation Consider the following differential equations:
dy
dx = ex (6)
© NCERT
not to be republished
DIFFERENTIAL EQUATIONS
381
2
d y2
y
dx
+
= 0 |
1 | 4434-4437 | Order of a differential equation
Order of a differential equation is defined as the order of the highest order derivative of
the dependent variable with respect to the independent variable involved in the given
differential equation Consider the following differential equations:
dy
dx = ex (6)
© NCERT
not to be republished
DIFFERENTIAL EQUATIONS
381
2
d y2
y
dx
+
= 0 (7)
3
3
2
2
3
2
d y
d y
x
dx
dx
⎛
⎞
⎛
⎞
+
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
= 0 |
1 | 4435-4438 | Consider the following differential equations:
dy
dx = ex (6)
© NCERT
not to be republished
DIFFERENTIAL EQUATIONS
381
2
d y2
y
dx
+
= 0 (7)
3
3
2
2
3
2
d y
d y
x
dx
dx
⎛
⎞
⎛
⎞
+
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
= 0 (8)
The equations (6), (7) and (8) involve the highest derivative of first, second and
third order respectively |
1 | 4436-4439 | (6)
© NCERT
not to be republished
DIFFERENTIAL EQUATIONS
381
2
d y2
y
dx
+
= 0 (7)
3
3
2
2
3
2
d y
d y
x
dx
dx
⎛
⎞
⎛
⎞
+
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
= 0 (8)
The equations (6), (7) and (8) involve the highest derivative of first, second and
third order respectively Therefore, the order of these equations are 1, 2 and 3 respectively |
1 | 4437-4440 | (7)
3
3
2
2
3
2
d y
d y
x
dx
dx
⎛
⎞
⎛
⎞
+
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
= 0 (8)
The equations (6), (7) and (8) involve the highest derivative of first, second and
third order respectively Therefore, the order of these equations are 1, 2 and 3 respectively 9 |
1 | 4438-4441 | (8)
The equations (6), (7) and (8) involve the highest derivative of first, second and
third order respectively Therefore, the order of these equations are 1, 2 and 3 respectively 9 2 |
1 | 4439-4442 | Therefore, the order of these equations are 1, 2 and 3 respectively 9 2 2 Degree of a differential equation
To study the degree of a differential equation, the key point is that the differential
equation must be a polynomial equation in derivatives, i |
1 | 4440-4443 | 9 2 2 Degree of a differential equation
To study the degree of a differential equation, the key point is that the differential
equation must be a polynomial equation in derivatives, i e |
1 | 4441-4444 | 2 2 Degree of a differential equation
To study the degree of a differential equation, the key point is that the differential
equation must be a polynomial equation in derivatives, i e , y′, y″, y″′ etc |
1 | 4442-4445 | 2 Degree of a differential equation
To study the degree of a differential equation, the key point is that the differential
equation must be a polynomial equation in derivatives, i e , y′, y″, y″′ etc Consider the
following differential equations:
2
3
2
3
2
2
d y
d y
dy
y
dx
dx
dx
⎛
⎞
+
−
+
⎜
⎟
⎝
⎠
= 0 |
1 | 4443-4446 | e , y′, y″, y″′ etc Consider the
following differential equations:
2
3
2
3
2
2
d y
d y
dy
y
dx
dx
dx
⎛
⎞
+
−
+
⎜
⎟
⎝
⎠
= 0 (9)
2
sin2
dy
dy
y
dx
dx
⎛
⎞
⎛
⎞
+
−
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
= 0 |
1 | 4444-4447 | , y′, y″, y″′ etc Consider the
following differential equations:
2
3
2
3
2
2
d y
d y
dy
y
dx
dx
dx
⎛
⎞
+
−
+
⎜
⎟
⎝
⎠
= 0 (9)
2
sin2
dy
dy
y
dx
dx
⎛
⎞
⎛
⎞
+
−
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
= 0 (10)
sin
dy
dy
dx
⎛dx
⎞
+
⎜
⎟
⎝
⎠ = 0 |
1 | 4445-4448 | Consider the
following differential equations:
2
3
2
3
2
2
d y
d y
dy
y
dx
dx
dx
⎛
⎞
+
−
+
⎜
⎟
⎝
⎠
= 0 (9)
2
sin2
dy
dy
y
dx
dx
⎛
⎞
⎛
⎞
+
−
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
= 0 (10)
sin
dy
dy
dx
⎛dx
⎞
+
⎜
⎟
⎝
⎠ = 0 (11)
We observe that equation (9) is a polynomial equation in y″′, y″ and y′, equation (10)
is a polynomial equation in y′ (not a polynomial in y though) |
1 | 4446-4449 | (9)
2
sin2
dy
dy
y
dx
dx
⎛
⎞
⎛
⎞
+
−
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
= 0 (10)
sin
dy
dy
dx
⎛dx
⎞
+
⎜
⎟
⎝
⎠ = 0 (11)
We observe that equation (9) is a polynomial equation in y″′, y″ and y′, equation (10)
is a polynomial equation in y′ (not a polynomial in y though) Degree of such differential
equations can be defined |
1 | 4447-4450 | (10)
sin
dy
dy
dx
⎛dx
⎞
+
⎜
⎟
⎝
⎠ = 0 (11)
We observe that equation (9) is a polynomial equation in y″′, y″ and y′, equation (10)
is a polynomial equation in y′ (not a polynomial in y though) Degree of such differential
equations can be defined But equation (11) is not a polynomial equation in y′ and
degree of such a differential equation can not be defined |
1 | 4448-4451 | (11)
We observe that equation (9) is a polynomial equation in y″′, y″ and y′, equation (10)
is a polynomial equation in y′ (not a polynomial in y though) Degree of such differential
equations can be defined But equation (11) is not a polynomial equation in y′ and
degree of such a differential equation can not be defined By the degree of a differential equation, when it is a polynomial equation in
derivatives, we mean the highest power (positive integral index) of the highest order
derivative involved in the given differential equation |
1 | 4449-4452 | Degree of such differential
equations can be defined But equation (11) is not a polynomial equation in y′ and
degree of such a differential equation can not be defined By the degree of a differential equation, when it is a polynomial equation in
derivatives, we mean the highest power (positive integral index) of the highest order
derivative involved in the given differential equation In view of the above definition, one may observe that differential equations (6), (7),
(8) and (9) each are of degree one, equation (10) is of degree two while the degree of
differential equation (11) is not defined |
1 | 4450-4453 | But equation (11) is not a polynomial equation in y′ and
degree of such a differential equation can not be defined By the degree of a differential equation, when it is a polynomial equation in
derivatives, we mean the highest power (positive integral index) of the highest order
derivative involved in the given differential equation In view of the above definition, one may observe that differential equations (6), (7),
(8) and (9) each are of degree one, equation (10) is of degree two while the degree of
differential equation (11) is not defined �Note Order and degree (if defined) of a differential equation are always
positive integers |
1 | 4451-4454 | By the degree of a differential equation, when it is a polynomial equation in
derivatives, we mean the highest power (positive integral index) of the highest order
derivative involved in the given differential equation In view of the above definition, one may observe that differential equations (6), (7),
(8) and (9) each are of degree one, equation (10) is of degree two while the degree of
differential equation (11) is not defined �Note Order and degree (if defined) of a differential equation are always
positive integers © NCERT
not to be republished
MATHEMATICS
382
Example 1 Find the order and degree, if defined, of each of the following differential
equations:
(i)
cos
0
dy
x
dx −
=
(ii)
2
2
2
0
d y
dy
dy
xy
x
y
dx
dx
dx
⎛
⎞
+
−
=
⎜
⎟
⎝
⎠
(iii)
2
0
y
y
y
e ′
′′′ +
+
=
Solution
(i)
The highest order derivative present in the differential equation is dy
dx , so its
order is one |
1 | 4452-4455 | In view of the above definition, one may observe that differential equations (6), (7),
(8) and (9) each are of degree one, equation (10) is of degree two while the degree of
differential equation (11) is not defined �Note Order and degree (if defined) of a differential equation are always
positive integers © NCERT
not to be republished
MATHEMATICS
382
Example 1 Find the order and degree, if defined, of each of the following differential
equations:
(i)
cos
0
dy
x
dx −
=
(ii)
2
2
2
0
d y
dy
dy
xy
x
y
dx
dx
dx
⎛
⎞
+
−
=
⎜
⎟
⎝
⎠
(iii)
2
0
y
y
y
e ′
′′′ +
+
=
Solution
(i)
The highest order derivative present in the differential equation is dy
dx , so its
order is one It is a polynomial equation in y′ and the highest power raised to dy
dx
is one, so its degree is one |
1 | 4453-4456 | �Note Order and degree (if defined) of a differential equation are always
positive integers © NCERT
not to be republished
MATHEMATICS
382
Example 1 Find the order and degree, if defined, of each of the following differential
equations:
(i)
cos
0
dy
x
dx −
=
(ii)
2
2
2
0
d y
dy
dy
xy
x
y
dx
dx
dx
⎛
⎞
+
−
=
⎜
⎟
⎝
⎠
(iii)
2
0
y
y
y
e ′
′′′ +
+
=
Solution
(i)
The highest order derivative present in the differential equation is dy
dx , so its
order is one It is a polynomial equation in y′ and the highest power raised to dy
dx
is one, so its degree is one (ii)
The highest order derivative present in the given differential equation is
2
d y2
dx
, so
its order is two |
1 | 4454-4457 | © NCERT
not to be republished
MATHEMATICS
382
Example 1 Find the order and degree, if defined, of each of the following differential
equations:
(i)
cos
0
dy
x
dx −
=
(ii)
2
2
2
0
d y
dy
dy
xy
x
y
dx
dx
dx
⎛
⎞
+
−
=
⎜
⎟
⎝
⎠
(iii)
2
0
y
y
y
e ′
′′′ +
+
=
Solution
(i)
The highest order derivative present in the differential equation is dy
dx , so its
order is one It is a polynomial equation in y′ and the highest power raised to dy
dx
is one, so its degree is one (ii)
The highest order derivative present in the given differential equation is
2
d y2
dx
, so
its order is two It is a polynomial equation in
2
d y2
dx
and dy
dx and the highest
power raised to
2
d y2
dx
is one, so its degree is one |
1 | 4455-4458 | It is a polynomial equation in y′ and the highest power raised to dy
dx
is one, so its degree is one (ii)
The highest order derivative present in the given differential equation is
2
d y2
dx
, so
its order is two It is a polynomial equation in
2
d y2
dx
and dy
dx and the highest
power raised to
2
d y2
dx
is one, so its degree is one (iii)
The highest order derivative present in the differential equation is y′′′, so its
order is three |
1 | 4456-4459 | (ii)
The highest order derivative present in the given differential equation is
2
d y2
dx
, so
its order is two It is a polynomial equation in
2
d y2
dx
and dy
dx and the highest
power raised to
2
d y2
dx
is one, so its degree is one (iii)
The highest order derivative present in the differential equation is y′′′, so its
order is three The given differential equation is not a polynomial equation in its
derivatives and so its degree is not defined |
1 | 4457-4460 | It is a polynomial equation in
2
d y2
dx
and dy
dx and the highest
power raised to
2
d y2
dx
is one, so its degree is one (iii)
The highest order derivative present in the differential equation is y′′′, so its
order is three The given differential equation is not a polynomial equation in its
derivatives and so its degree is not defined EXERCISE 9 |
1 | 4458-4461 | (iii)
The highest order derivative present in the differential equation is y′′′, so its
order is three The given differential equation is not a polynomial equation in its
derivatives and so its degree is not defined EXERCISE 9 1
Determine order and degree (if defined) of differential equations given in Exercises
1 to 10 |
1 | 4459-4462 | The given differential equation is not a polynomial equation in its
derivatives and so its degree is not defined EXERCISE 9 1
Determine order and degree (if defined) of differential equations given in Exercises
1 to 10 1 |
1 | 4460-4463 | EXERCISE 9 1
Determine order and degree (if defined) of differential equations given in Exercises
1 to 10 1 4
4
sin(
)
0
d y
y
dx
′′′
+
=
2 |
1 | 4461-4464 | 1
Determine order and degree (if defined) of differential equations given in Exercises
1 to 10 1 4
4
sin(
)
0
d y
y
dx
′′′
+
=
2 y′ + 5y = 0
3 |
1 | 4462-4465 | 1 4
4
sin(
)
0
d y
y
dx
′′′
+
=
2 y′ + 5y = 0
3 4
2
2
3
0
ds
sd s
dt
dt
⎛
⎞ +
=
⎜
⎟
⎝
⎠
4 |
1 | 4463-4466 | 4
4
sin(
)
0
d y
y
dx
′′′
+
=
2 y′ + 5y = 0
3 4
2
2
3
0
ds
sd s
dt
dt
⎛
⎞ +
=
⎜
⎟
⎝
⎠
4 2
2
2
cos
0
d y
dy
dx
dx
⎛
⎞
⎛
⎞
+
=
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
5 |
1 | 4464-4467 | y′ + 5y = 0
3 4
2
2
3
0
ds
sd s
dt
dt
⎛
⎞ +
=
⎜
⎟
⎝
⎠
4 2
2
2
cos
0
d y
dy
dx
dx
⎛
⎞
⎛
⎞
+
=
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
5 2
2
cos3
sin3
d y
x
x
dx
=
+
6 |
1 | 4465-4468 | 4
2
2
3
0
ds
sd s
dt
dt
⎛
⎞ +
=
⎜
⎟
⎝
⎠
4 2
2
2
cos
0
d y
dy
dx
dx
⎛
⎞
⎛
⎞
+
=
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
5 2
2
cos3
sin3
d y
x
x
dx
=
+
6 2
(
y′′′)
+ (y″)3 + (y′)4 + y5 = 0
7 |
1 | 4466-4469 | 2
2
2
cos
0
d y
dy
dx
dx
⎛
⎞
⎛
⎞
+
=
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
5 2
2
cos3
sin3
d y
x
x
dx
=
+
6 2
(
y′′′)
+ (y″)3 + (y′)4 + y5 = 0
7 y′′′ + 2y″ + y′ = 0
© NCERT
not to be republished
DIFFERENTIAL EQUATIONS
383
8 |
1 | 4467-4470 | 2
2
cos3
sin3
d y
x
x
dx
=
+
6 2
(
y′′′)
+ (y″)3 + (y′)4 + y5 = 0
7 y′′′ + 2y″ + y′ = 0
© NCERT
not to be republished
DIFFERENTIAL EQUATIONS
383
8 y′ + y = ex
9 |
1 | 4468-4471 | 2
(
y′′′)
+ (y″)3 + (y′)4 + y5 = 0
7 y′′′ + 2y″ + y′ = 0
© NCERT
not to be republished
DIFFERENTIAL EQUATIONS
383
8 y′ + y = ex
9 y″ + (y′)2 + 2y = 0 10 |
1 | 4469-4472 | y′′′ + 2y″ + y′ = 0
© NCERT
not to be republished
DIFFERENTIAL EQUATIONS
383
8 y′ + y = ex
9 y″ + (y′)2 + 2y = 0 10 y″ + 2y′ + sin y = 0
11 |
1 | 4470-4473 | y′ + y = ex
9 y″ + (y′)2 + 2y = 0 10 y″ + 2y′ + sin y = 0
11 The degree of the differential equation
3
2
2
2
sin
1
0
d y
dy
dy
dx
dx
dx
⎛
⎞
⎛
⎞
⎛
⎞
+
+
+ =
⎜
⎟
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
⎝
⎠
is
(A) 3
(B) 2
(C) 1
(D) not defined
12 |
1 | 4471-4474 | y″ + (y′)2 + 2y = 0 10 y″ + 2y′ + sin y = 0
11 The degree of the differential equation
3
2
2
2
sin
1
0
d y
dy
dy
dx
dx
dx
⎛
⎞
⎛
⎞
⎛
⎞
+
+
+ =
⎜
⎟
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
⎝
⎠
is
(A) 3
(B) 2
(C) 1
(D) not defined
12 The order of the differential equation
2
2
2
2
3
0
d y
dy
x
y
dx
dx
−
+
=
is
(A) 2
(B) 1
(C) 0
(D) not defined
9 |
1 | 4472-4475 | y″ + 2y′ + sin y = 0
11 The degree of the differential equation
3
2
2
2
sin
1
0
d y
dy
dy
dx
dx
dx
⎛
⎞
⎛
⎞
⎛
⎞
+
+
+ =
⎜
⎟
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
⎝
⎠
is
(A) 3
(B) 2
(C) 1
(D) not defined
12 The order of the differential equation
2
2
2
2
3
0
d y
dy
x
y
dx
dx
−
+
=
is
(A) 2
(B) 1
(C) 0
(D) not defined
9 3 |
1 | 4473-4476 | The degree of the differential equation
3
2
2
2
sin
1
0
d y
dy
dy
dx
dx
dx
⎛
⎞
⎛
⎞
⎛
⎞
+
+
+ =
⎜
⎟
⎜
⎟
⎜
⎟
⎝
⎠
⎝
⎠
⎝
⎠
is
(A) 3
(B) 2
(C) 1
(D) not defined
12 The order of the differential equation
2
2
2
2
3
0
d y
dy
x
y
dx
dx
−
+
=
is
(A) 2
(B) 1
(C) 0
(D) not defined
9 3 General and Particular Solutions of a Differential Equation
In earlier Classes, we have solved the equations of the type:
x2 + 1 = 0 |
1 | 4474-4477 | The order of the differential equation
2
2
2
2
3
0
d y
dy
x
y
dx
dx
−
+
=
is
(A) 2
(B) 1
(C) 0
(D) not defined
9 3 General and Particular Solutions of a Differential Equation
In earlier Classes, we have solved the equations of the type:
x2 + 1 = 0 (1)
sin2 x – cos x = 0 |
1 | 4475-4478 | 3 General and Particular Solutions of a Differential Equation
In earlier Classes, we have solved the equations of the type:
x2 + 1 = 0 (1)
sin2 x – cos x = 0 (2)
Solution of equations (1) and (2) are numbers, real or complex, that will satisfy the
given equation i |
1 | 4476-4479 | General and Particular Solutions of a Differential Equation
In earlier Classes, we have solved the equations of the type:
x2 + 1 = 0 (1)
sin2 x – cos x = 0 (2)
Solution of equations (1) and (2) are numbers, real or complex, that will satisfy the
given equation i e |
1 | 4477-4480 | (1)
sin2 x – cos x = 0 (2)
Solution of equations (1) and (2) are numbers, real or complex, that will satisfy the
given equation i e , when that number is substituted for the unknown x in the given
equation, L |
1 | 4478-4481 | (2)
Solution of equations (1) and (2) are numbers, real or complex, that will satisfy the
given equation i e , when that number is substituted for the unknown x in the given
equation, L H |
1 | 4479-4482 | e , when that number is substituted for the unknown x in the given
equation, L H S |
1 | 4480-4483 | , when that number is substituted for the unknown x in the given
equation, L H S becomes equal to the R |
1 | 4481-4484 | H S becomes equal to the R H |
1 | 4482-4485 | S becomes equal to the R H S |
1 | 4483-4486 | becomes equal to the R H S Now consider the differential equation
2
2
0
d y
y
dx
+
= |
1 | 4484-4487 | H S Now consider the differential equation
2
2
0
d y
y
dx
+
= (3)
In contrast to the first two equations, the solution of this differential equation is a
function φ that will satisfy it i |
1 | 4485-4488 | S Now consider the differential equation
2
2
0
d y
y
dx
+
= (3)
In contrast to the first two equations, the solution of this differential equation is a
function φ that will satisfy it i e |
1 | 4486-4489 | Now consider the differential equation
2
2
0
d y
y
dx
+
= (3)
In contrast to the first two equations, the solution of this differential equation is a
function φ that will satisfy it i e , when the function φ is substituted for the unknown y
(dependent variable) in the given differential equation, L |
1 | 4487-4490 | (3)
In contrast to the first two equations, the solution of this differential equation is a
function φ that will satisfy it i e , when the function φ is substituted for the unknown y
(dependent variable) in the given differential equation, L H |
1 | 4488-4491 | e , when the function φ is substituted for the unknown y
(dependent variable) in the given differential equation, L H S |
1 | 4489-4492 | , when the function φ is substituted for the unknown y
(dependent variable) in the given differential equation, L H S becomes equal to R |
1 | 4490-4493 | H S becomes equal to R H |
1 | 4491-4494 | S becomes equal to R H S |
1 | 4492-4495 | becomes equal to R H S The curve y = φ (x) is called the solution curve (integral curve) of the given
differential equation |
1 | 4493-4496 | H S The curve y = φ (x) is called the solution curve (integral curve) of the given
differential equation Consider the function given by
y = φ (x) = a sin (x + b), |
1 | 4494-4497 | S The curve y = φ (x) is called the solution curve (integral curve) of the given
differential equation Consider the function given by
y = φ (x) = a sin (x + b), (4)
where a, b ∈ R |
1 | 4495-4498 | The curve y = φ (x) is called the solution curve (integral curve) of the given
differential equation Consider the function given by
y = φ (x) = a sin (x + b), (4)
where a, b ∈ R When this function and its derivative are substituted in equation (3),
L |
1 | 4496-4499 | Consider the function given by
y = φ (x) = a sin (x + b), (4)
where a, b ∈ R When this function and its derivative are substituted in equation (3),
L H |
1 | 4497-4500 | (4)
where a, b ∈ R When this function and its derivative are substituted in equation (3),
L H S |
1 | 4498-4501 | When this function and its derivative are substituted in equation (3),
L H S = R |
1 | 4499-4502 | H S = R H |
1 | 4500-4503 | S = R H S |
1 | 4501-4504 | = R H S So it is a solution of the differential equation (3) |
1 | 4502-4505 | H S So it is a solution of the differential equation (3) Let a and b be given some particular values say a = 2 and
4
b
=π
, then we get a
function
y = φ1(x) = 2sin
4
x
π
⎛
⎞
+
⎜
⎟
⎝
⎠ |
1 | 4503-4506 | S So it is a solution of the differential equation (3) Let a and b be given some particular values say a = 2 and
4
b
=π
, then we get a
function
y = φ1(x) = 2sin
4
x
π
⎛
⎞
+
⎜
⎟
⎝
⎠ (5)
When this function and its derivative are substituted in equation (3) again
L |
1 | 4504-4507 | So it is a solution of the differential equation (3) Let a and b be given some particular values say a = 2 and
4
b
=π
, then we get a
function
y = φ1(x) = 2sin
4
x
π
⎛
⎞
+
⎜
⎟
⎝
⎠ (5)
When this function and its derivative are substituted in equation (3) again
L H |
1 | 4505-4508 | Let a and b be given some particular values say a = 2 and
4
b
=π
, then we get a
function
y = φ1(x) = 2sin
4
x
π
⎛
⎞
+
⎜
⎟
⎝
⎠ (5)
When this function and its derivative are substituted in equation (3) again
L H S |
1 | 4506-4509 | (5)
When this function and its derivative are substituted in equation (3) again
L H S = R |
1 | 4507-4510 | H S = R H |
1 | 4508-4511 | S = R H S |
1 | 4509-4512 | = R H S Therefore φ1 is also a solution of equation (3) |
1 | 4510-4513 | H S Therefore φ1 is also a solution of equation (3) © NCERT
not to be republished
MATHEMATICS
384
Function φ consists of two arbitrary constants (parameters) a, b and it is called
general solution of the given differential equation |
1 | 4511-4514 | S Therefore φ1 is also a solution of equation (3) © NCERT
not to be republished
MATHEMATICS
384
Function φ consists of two arbitrary constants (parameters) a, b and it is called
general solution of the given differential equation Whereas function φ1 contains no
arbitrary constants but only the particular values of the parameters a and b and hence
is called a particular solution of the given differential equation |
1 | 4512-4515 | Therefore φ1 is also a solution of equation (3) © NCERT
not to be republished
MATHEMATICS
384
Function φ consists of two arbitrary constants (parameters) a, b and it is called
general solution of the given differential equation Whereas function φ1 contains no
arbitrary constants but only the particular values of the parameters a and b and hence
is called a particular solution of the given differential equation The solution which contains arbitrary constants is called the general solution
(primitive) of the differential equation |
1 | 4513-4516 | © NCERT
not to be republished
MATHEMATICS
384
Function φ consists of two arbitrary constants (parameters) a, b and it is called
general solution of the given differential equation Whereas function φ1 contains no
arbitrary constants but only the particular values of the parameters a and b and hence
is called a particular solution of the given differential equation The solution which contains arbitrary constants is called the general solution
(primitive) of the differential equation The solution free from arbitrary constants i |
1 | 4514-4517 | Whereas function φ1 contains no
arbitrary constants but only the particular values of the parameters a and b and hence
is called a particular solution of the given differential equation The solution which contains arbitrary constants is called the general solution
(primitive) of the differential equation The solution free from arbitrary constants i e |
1 | 4515-4518 | The solution which contains arbitrary constants is called the general solution
(primitive) of the differential equation The solution free from arbitrary constants i e , the solution obtained from the general
solution by giving particular values to the arbitrary constants is called a particular
solution of the differential equation |
1 | 4516-4519 | The solution free from arbitrary constants i e , the solution obtained from the general
solution by giving particular values to the arbitrary constants is called a particular
solution of the differential equation Example 2 Verify that the function y = e– 3x is a solution of the differential equation
2
2
6
0
d y
dy
y
dx
dx
+
−
=
Solution Given function is y = e– 3x |
1 | 4517-4520 | e , the solution obtained from the general
solution by giving particular values to the arbitrary constants is called a particular
solution of the differential equation Example 2 Verify that the function y = e– 3x is a solution of the differential equation
2
2
6
0
d y
dy
y
dx
dx
+
−
=
Solution Given function is y = e– 3x Differentiating both sides of equation with respect
to x , we get
3
3
x
dy
e
dx
−
= − |
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