# 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

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.

(A)

(B)

(C)

(D)

Ans. D