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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