NCERT Solutions class 12 Maths Exercise 9.5



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NCERT Solutions class 12 Maths Exercise 9.5 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.

Download NCERT solutions for Differential Equations  as PDF.

NCERT Solutions class 12 Maths Exercise 9.5

NCERT Solutions class 12 Maths Differential Equations

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)

 

 

 

 

   [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

 

=

 

 

 

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 , =

Multiplying within logs by  in L.H.S.,  =


NCERT Solutions class 12 Maths Exercise 9.5

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,

 

 

 

  =

 

 

 

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 =


NCERT Solutions class 12 Maths Exercise 9.5

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)

 

 

 

 

  ……….(ii)

Therefore the given differential equation is homogeneous because each coefficient of  and  is same i.e., degree 2.

Putting

 

 

Putting these values of  and  in eq. (ii), we have

 

 

 

 

  [Separating variables]

Integrating both sides,

 

 

Now putting  

 

 

 

  ……….(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.

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 i.e., degree 2.

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

 

 

 


NCERT Solutions class 12 Maths Exercise 9.5

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.


NCERT Solutions class 12 Maths Exercise 9.5

17. Which of the following is a homogeneous differential equation:

(A)  

(B)  

(C)  

(D)  

Ans. Out of the given four options, option (D) is the only option in which all coefficients of  and  are of same degree i.e., 2. It may be noted that  is a term of second degree.

Hence differential equation in option (D) is Homogeneous differential equation.

NCERT Solutions class 12 Maths Exercise 9.5

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