NCERT Solutions class 12 Maths Exercise 9.5

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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.,  =

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

=

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)

……….(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

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. 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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2 thoughts on “NCERT Solutions class 12 Maths Exercise 9.5”

1. In Q3 it should be 1+v² not 1-v²

2. Nice guide