NCERT Solutions class 12 Maths Exercise 9.5 (Ex 9.5) Chapter 9 Differential Equations

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.   
 
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 
  
 
Putting value of   and   in eq. (ii),
 
 
 
   [Separating variables]
Integrating both sides,
=> 
 
 
 
 
 
Putting    
 
 
 
 
 
   where C = 

2.   

 
 

Ans. Given: Differential equation  
 
   ……….(i)
Therefore, eq. (i) is homogeneous.
Putting 
  
 
Putting value of   and   in eq. (i)
=> 
 
 
   [Separating variables]
Integrating both sides,
=> 
 
Putting  ,
=> 
 

3.   

 
 

Ans. Given: Differential equation   ……….(i) 
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+vv+v21v=>xdvdx=1+v−v+v21−v
=>xdvdx=1+v21v=>xdvdx=1+v21−v
 
    [Separating variables]
Integrating both sides,
=> 
 
 
 
Putting  ,
=>
 
 
 
 
 
 

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

 
 xdvdx=v212v22vxdvdx=v2−1−2v22v
 
 
 
Integrating both sides,
=> 
 
 
 
Put  ,
=> 
 
 
 

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

 
 
   [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  
  
 
 
[Dividing by  ]
 
Therefore given differential equation is homogeneous.
Putting 
 
 
Putting these values of   and   in eq. (i), we get
=> 
 
 
Integrating both sides,
 
 
Putting   ,

 
 

7.   

 
 

Ans. Given: Differential equation  
  
   ……….(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.   

 
 

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,
=> 
 
 
 
   [putting  ]
 
  where 
 

9.   

 
 

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,
 
 
 
 
    where C = 
 
   [Putting  ]
 

10.   

 
 

Ans. Given: Differential equation  
   [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   
 

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

 
 
 
 
   [Separating variables]
Integrating both sides,
=> 
 
 
Now putting 
 
 
 
12log(y2+x2)12×2logx+tan1yx=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)+2tan1yx=log2+π4log(y2+x2)+2tan−1yx=log2+π4

12.   when 

 
 

Ans. Given: Differential equation  
 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 

 
 

Ans. Given: Differential equation  
 
   =    ……….(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 

 
 

Ans. Given: Differential equation  
    ………(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 

 
 

Ans. Given: Differential equation  ……….(i) 
 
 
    ……….(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)   
 

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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