NCERT Solutions for Class 12 Maths Exercise 9.5 Chapter 9 Differential Equations – FREE PDF Download
Free PDF download of NCERT Solutions for Class 12 Maths Chapter 9 Exercise 9.5 (Ex 9.5) and all chapter exercises at one place prepared by expert teacher as per NCERT (CBSE) books guidelines. Class 12 Maths Chapter 9 Differential Equations Exercise 9.5 Questions with Solutions to help you to revise complete Syllabus and Score More marks.
NCERT Solutions for Class 12 Maths Chapter 9 Differential Equations (Ex 9.5) 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.
![](https://cdn1.coolgyan.org/statics/12/maths/ncert/Ex9.5/image001.png)
Here degree of each coefficients of and
is same therefore, it is homogenous.
…..(ii)
F
,
therefore the given differential equation is homogeneous.
Putting
Putting value of and
in eq. (ii),
[Separating variables]
Integrating both sides,
=>
Putting
where C =
2.
![](https://cdn1.coolgyan.org/statics/12/maths/ncert/Ex9.5/image034.png)
……….(i)
Therefore, eq. (i) is homogeneous.
Putting
Putting value of and
in eq. (i)
=>
[Separating variables]
Integrating both sides,
=>
Putting ,
=>
3.
![](https://cdn1.coolgyan.org/statics/12/maths/ncert/Ex9.5/image047.png)
This given equation is homogeneous because each coefficients of and
is of degree 1.
Putting
….(ii)
Putting value of and
in eq. (ii)
=>xdvdx=1+v−v+v21−v=>xdvdx=1+v−v+v21−v
=>xdvdx=1+v21−v=>xdvdx=1+v21−v
[Separating variables]
Integrating both sides,
=>
Putting ,
=>
4.
![](https://cdn1.coolgyan.org/statics/12/maths/ncert/Ex9.5/image068.png)
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
xdvdx=v2−1−2v22vxdvdx=v2−1−2v22v
Integrating both sides,
=>
Put ,
=>
5.
![](https://cdn1.coolgyan.org/statics/12/maths/ncert/Ex9.5/image087.png)
=
……….(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
[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.
![](https://cdn1.coolgyan.org/statics/12/maths/ncert/Ex9.5/image103.png)
[Dividing by ]
Therefore given differential equation is homogeneous.
Putting
Putting these values of and
in eq. (i), we get
=>
Integrating both sides,
Putting ,
7.
![](https://cdn1.coolgyan.org/statics/12/maths/ncert/Ex9.5/image116.png)
……….(i)
Therefore, the given differential equation is homogeneous.
Putting
Putting these values of and
in eq. (i), we get
[Separating variables]
Integrating both sides,
log∣∣secvv∣∣=log|c|x2log|secvv|=log|c|x2
Putting
=> where C =
8.
![](https://cdn1.coolgyan.org/statics/12/maths/ncert/Ex9.5/image141.png)
=
……….(i)
Therefore, the given differential equation is homogeneous.
Putting
Putting these values of and
in eq. (i), we get
Integrating both sides,
=>
[putting
]
where
9.
![](https://cdn1.coolgyan.org/statics/12/maths/ncert/Ex9.5/image159.png)
……….(i)
Therefore, the given differential equation is homogeneous.
Putting
Putting these values of and
in eq. (i), we get
=
Integrating both sides,
where C =
[Putting
]
10.
![](https://cdn1.coolgyan.org/statics/12/maths/ncert/Ex9.5/image181.png)
[Dividing by
]
……….(i)
Therefore, it is a homogeneous.
Now putting
Putting these values of and
in eq. (i), we have
[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 solution satisfying the given condition
11. when
![](https://cdn1.coolgyan.org/statics/12/maths/ncert/Ex9.5/image205.png)
![](https://cdn1.coolgyan.org/statics/12/maths/ncert/Ex9.5/image204.png)
(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
[Separating variables]
Integrating both sides,
=>
Now putting
12log(y2+x2)−12×2logx+tan−1yx=−logx+c12log(y2+x2)−12×2logx+tan−1yx=−logx+c
……….(iii)
Now again given when
, therefore putting these values in eq. (iii),
Putting this value of in eq. (iii), we get
log(y2+x2)+2tan−1yx=log2+π4log(y2+x2)+2tan−1yx=log2+π4
12.
when ![](data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==)
![](https://cdn1.coolgyan.org/statics/12/maths/ncert/Ex9.5/image236.png)
x2dy=−(xy+y2)dxx2dy=−(xy+y2)dx
……….(i)
Therefore the given differential equation is homogeneous.
Putting
Putting these values of and
in eq. (i), we have
Integrating both sides,
Putting
where C =
……….(ii)
Now putting and
in eq. (ii), we get 1 = 3C
Putting value of C in eq. (ii),
13.
when ![](data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==)
![](https://cdn1.coolgyan.org/statics/12/maths/ncert/Ex9.5/image259.png)
=
……….(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),
Putting the value of in eq. (ii),
14.
when ![](data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==)
![](https://cdn1.coolgyan.org/statics/12/maths/ncert/Ex9.5/image278.png)
………(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),
Putting the value of in eq. (ii),
15.
when ![](data:image/gif;base64,R0lGODlhAQABAAAAACH5BAEKAAEALAAAAAABAAEAAAICTAEAOw==)
![](https://cdn1.coolgyan.org/statics/12/maths/ncert/Ex9.5/image294.png)
……….(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,
[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:
(A)
(B)
(C)
(D)
![](https://cdn1.coolgyan.org/statics/12/maths/ncert/Ex9.5/image317.png)
![](https://cdn1.coolgyan.org/statics/12/maths/ncert/Ex9.5/image318.png)
![](https://cdn1.coolgyan.org/statics/12/maths/ncert/Ex9.5/image319.png)
Therefore, option (C) is correct.
17. Which of the following is a homogeneous differential equation:
(A)
(B)
(C)
(D)