cbse class ncert solution chapter - 9 differential equations - … · 2020. 3. 12. · in each of...
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CBSE Class–12 Mathematics
NCERT solution Chapter - 9
Differential Equations - Exercise 9.5
In each of the following Questions 1 to 5, show that the differential equation is
homogenous and solve each of them:
1.
Ans. Given: Differential equation ……….(i)
Here degree of each coefficients of and is same therefore, it is homogenous.
…..(ii)
F ,
therefore the given differential equation is homogeneous.
Putting
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Putting value of and in eq. (ii),
[Separating variables]
Integrating both sides,
=>
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Putting
where C =
2.
Ans. Given: Differential equation
……….(i)
Therefore, eq. (i) is homogeneous.
Putting
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Putting value of and in eq. (i)
=>
[Separating variables]
Integrating both sides,
=>
Putting ,
=>
3.
Ans. Given: Differential equation ……….(i)
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This given equation is homogeneous because each coefficients of and is of degree 1.
Putting
….(ii)
Putting value of and in eq. (ii)
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[Separating variables]
Integrating both sides,
=>
Putting ,
=>
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4.
Ans. Given: Differential equation -
This equation is homogeneous because degree of each coefficient of and is same i.e.,
2
……….(ii)
Therefore, the given equation is homogeneous.
Put
Putting these values of and in eq. (ii), we get
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Integrating both sides,
=>
Put ,
=>
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5.
Ans. Given: Differential equation
= ……….(i)
Therefore, the given differential equation is homogeneous as all terms of and are of
same degree i.e., degree 2.
Putting
Putting these values of and in eq. (i), we get
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[Separating variables]
Integrating both sides,
=>
=
Putting ,
=> =
=> =
In each of the Questions 6 to 10, show that the given differential equation is
homogeneous and solve each of them:
6.
Ans. Given: Differential equation
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[Dividing by ]
Therefore given differential equation is homogeneous.
Putting
Putting these values of and in eq. (i), we get
=>
Integrating both sides,
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Putting ,
7.
Ans. Given: Differential equation
……….(i)
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Therefore, the given differential equation is homogeneous.
Putting
Putting these values of and in eq. (i), we get
[Separating variables]
Integrating both sides,
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Putting
=> where C =
8.
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Ans. Given: Differential equation
= ……….(i)
Therefore, the given differential equation is homogeneous.
Putting
Putting these values of and in eq. (i), we get
Integrating both sides,
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=>
[putting ]
where
9.
Ans. Given: Differential equation
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……….(i)
Therefore, the given differential equation is homogeneous.
Putting
Putting these values of and in eq. (i), we get
=
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Integrating both sides,
where C =
[Putting ]
10.
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Ans. Given: Differential equation
[Dividing by ]
……….(i)
Therefore, it is a homogeneous.
Now putting
Putting these values of and in eq. (i), we have
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[Separating variables]
Integrating both sides,
Now putting ,
C where C =
For each of the differential equations in Questions from 11 to 15, find the particular
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solution satisfying the given condition
11. when
Ans. Given: Differential equation when …..(i)
(x+y)dy+(x-y)dx=0
……….(ii)
Therefore the given differential equation is homogeneous because each coefficient of
and is same i.e., degree 1.
Putting
Putting these values of and in eq. (ii), we have
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[Separating variables]
Integrating both sides,
=>
Now putting
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……….(iii)
Now again given when , therefore putting these values in eq. (iii),
Putting this value of in eq. (iii), we get
12. when
Ans. Given: Differential equation
……….(i)
Therefore the given differential equation is homogeneous.
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Putting
Putting these values of and in eq. (i), we have
Integrating both sides,
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Putting
where C = ……….(ii)
Now putting and in eq. (ii), we get 1 = 3C
Putting value of C in eq. (ii),
13. when
Ans. Given: Differential equation
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= ……….(i)
Therefore, the given differential equation is homogeneous.
Putting
Putting these values of and in eq. (i), we have
[Separating variables]
Integrating both sides,
[Putting ] ……….(ii)
Now putting in eq. (ii),
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Putting the value of in eq. (ii),
14. when
Ans. Given: Differential equation
………(i)
Therefore, the given differential equation is homogeneous.
Putting
Putting these values of and in eq. (i), we have
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[Separating variables]
Integrating both sides,
[Putting ] ……….(ii)
Now putting in eq. (ii),
Putting the value of in eq. (ii),
15. when
Ans. Given: Differential equation ……….(i)
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……….(ii)
Therefore the given differential equation is homogeneous because each coefficient of
and is same.
Putting
Putting these values of and in eq. (ii), we have
[Separating variables]
Integrating both sides,
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[Putting ]
Now putting in ,
Again putting , in , we get
Choose the correct answer:
16. A homogeneous differential equation of the form can be solved by
making the substitution:
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(A)
(B)
(C)
(D)
Ans. We know that a homogeneous differential equation of the form can be
solved by the substitution i.e.,
Therefore, option (C) is correct.
17. Which of the following is a homogeneous differential equation:
(A)
(B)
(C)
(D)
Ans. D
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