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),

 

 

 

  Separatingvariables

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)

=> 

 

 

  Separatingvariables

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

 

   Separatingvariables

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

 

 

  Separatingvariables

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

 

 

 

 

  Separatingvariables

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

 

 

 

  Separatingvariables

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

 

 

 

 

  Separatingvariables

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

 

  Separatingvariables

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

 

 

  Separatingvariables

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

 

 

  Separatingvariables

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