## NCERT Solutions for Class 12 Maths Exercise 9.5 Chapter 9 Differential Equations â€“ 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.Â**

**Â**

**Ans.Â**Given: Differential equation Â Â â€¦â€¦â€¦.(i)

Here degree of each coefficients ofÂ Â andÂ Â is same therefore, it is homogenous.

Â Â

Â

Â Â â€¦..(ii)

Â F Â ,

therefore the given differential equation is homogeneous.

PuttingÂ

Â Â

Â

Putting value ofÂ Â andÂ Â in eq. (ii),

Â

Â

Â

Â Â [Separating variables]

Integrating both sides,

=>Â

Â

Â

Â

Â

Â

PuttingÂ Â Â

Â

Â

Â

Â

Â

Â Â where C =Â

###
**2.Â ****
****Â **

**Ans.Â**Given: Differential equation

Â

Â Â â€¦â€¦â€¦.(i)

Therefore, eq. (i) is homogeneous.

PuttingÂ

Â Â

Â

Putting value ofÂ Â andÂ Â in eq. (i)

=>Â

Â

Â

Â Â [Separating variables]

Integrating both sides,

=>Â

Â

PuttingÂ ,

=>Â

Â

###
**3.Â ****
****Â **

**Ans.Â**Given: Differential equation Â Â â€¦â€¦â€¦.(i)

This given equation is homogeneous because each coefficients ofÂ Â andÂ Â is of degree 1.

Â Â

Â

Â

PuttingÂ

Â Â

Â Â â€¦.(ii)

Putting value ofÂ Â andÂ Â in eq. (ii)

Â

=>xdvdx=1+vâˆ’v+v21âˆ’v=>xdvdx=1+vâˆ’v+v21âˆ’v

=>xdvdx=1+v21âˆ’v=>xdvdx=1+v21âˆ’v

Â

Â Â [Separating variables]

Integrating both sides,

=>Â

Â

Â

Â

PuttingÂ ,

=>

Â

Â

Â

Â

Â

Â

###
**4.Â ****
****Â **

**Ans.Â**Given: Differential equation Â â€“

This equation is homogeneous because degree of each coefficient ofÂ Â andÂ Â is same i.e., 2

Â Â

Â

Â Â â€¦â€¦â€¦.(ii)

Therefore, the given equation is homogeneous.

PutÂ

Â

Â

Putting these values ofÂ Â andÂ Â in eq. (ii), we get

Â

Â xdvdx=v2âˆ’1âˆ’2v22vxdvdx=v2âˆ’1âˆ’2v22v

Â

Â

Â

Integrating both sides,

=>Â

Â

Â

Â

PutÂ ,

=>Â

Â

Â

Â

###
**5.Â ****
****Â **

**Ans.Â**Given: Differential equation

Â Â

Â Â =Â Â Â â€¦â€¦â€¦.(i)

Therefore, the given differential equation is homogeneous as all terms ofÂ Â andÂ Â are of same degree i.e., degree 2.

PuttingÂ

Â

Â

Putting these values ofÂ Â andÂ Â in eq. (i), we get

Â

Â

Â Â [Separating variables]

Integrating both sides,

=>Â

Â

Â Â =Â

PuttingÂ ,

=>Â Â =Â

=>Â Â Â =Â

**In each of the Questions 6 to 10, show that the given differential equation is homogeneous and solve each of them:**

**6.Â ****
****Â **

**Ans.Â**Given: Differential equation

Â Â

Â

Â

[Dividing byÂ ]

Â

Therefore given differential equation is homogeneous.

PuttingÂ

Â

Â

Putting these values ofÂ Â andÂ Â in eq. (i), we get

=>Â

Â

Â

Integrating both sides,

Â

Â

PuttingÂ Â ,

Â

Â

###
**7.Â ****
****Â **

**Ans.Â**Given: Differential equation

Â Â

Â Â â€¦â€¦â€¦.(i)

Therefore, the given differential equation is homogeneous.

PuttingÂ

Â

Â

Putting these values ofÂ Â andÂ Â in eq. (i), we get

Â

Â

Â

Â

Â Â [Separating variables]

Integrating both sides,

Â Â

Â

Â

Â

logâˆ£âˆ£secvvâˆ£âˆ£=log|c|x2log|secvv|=log|c|x2

Â

Â

PuttingÂ

=>Â Â Â Â Â Â Â Â Â Â where C =Â

Â

Â

Â

Â

###
**8.Â ****
****Â **

**Ans.Â**Given: Differential equation

Â Â

Â Â Â =Â Â â€¦â€¦â€¦.(i)

Therefore, the given differential equation is homogeneous.

PuttingÂ

Â

Â

Putting these values ofÂ Â andÂ Â in eq. (i), we get

Â Â

Â

Â

Â

Integrating both sides,

=>Â

Â

Â

Â

Â Â [puttingÂ ]

Â

Â whereÂ

Â

###
**9.Â ****
****Â **

**Ans.Â**Given: Differential equation

Â

Â

Â Â â€¦â€¦â€¦.(i)

Therefore, the given differential equation is homogeneous.

PuttingÂ

Â

Â

Putting these values ofÂ Â andÂ Â in eq. (i), we get

Â

Â Â

=Â

Â

Â

Â

Â

Â

Â

Integrating both sides,

Â

Â

Â

Â

Â Â where C =Â

Â

Â Â [PuttingÂ ]

Â

###
**10.Â ****
****Â **

**Ans.Â**Given: Differential equation

Â Â [Dividing byÂ ]

Â Â

Â Â Â â€¦â€¦â€¦.(i)

Therefore, it is a homogeneous.

Now puttingÂ

Â

Â

Putting these values ofÂ Â andÂ Â in eq. (i), we have

Â

Â

Â

Â Â [Separating variables]

Integrating both sides,

Â

Now puttingÂ ,

Â

Â

Â

Â Â C where C =Â

**For each of the differential equations in Questions from 11 to 15, find the particular solution satisfying the given condition**

**11.Â ****
****Â whenÂ ****
****Â **

**Ans.Â**Given: Differential equation Â when Â Â â€¦..(i)

Â (x+y)dy+(x-y)dx=0

Â

Â

Â

Â Â â€¦â€¦â€¦.(ii)

Therefore the given differential equation is homogeneous because each coefficient ofÂ Â andÂ Â is same i.e., degree 1.

PuttingÂ

Â

Â

Putting these values ofÂ Â andÂ Â in eq. (ii), we have

Â

Â

Â

Â

Â Â [Separating variables]

Integrating both sides,

=>Â

Â

Â

Now puttingÂ

Â

Â

Â

12log(y2+x2)âˆ’12Ã—2logx+tanâˆ’1yx=âˆ’logx+c12log(y2+x2)âˆ’12Ã—2logx+tanâˆ’1yx=âˆ’logx+c

Â Â â€¦â€¦â€¦.(iii)

Now again givenÂ Â whenÂ , therefore putting these values in eq. (iii),

Â

Putting this value ofÂ Â in eq. (iii), we get

log(y2+x2)+2tanâˆ’1yx=log2+Ï€4log(y2+x2)+2tanâˆ’1yx=log2+Ï€4

###
**12.Â ****
****Â whenÂ ****
**

**Ans.Â**Given: Differential equation

Â x2dy=âˆ’(xy+y2)dxx2dy=âˆ’(xy+y2)dx

Â

Â Â â€¦â€¦â€¦.(i)

Therefore the given differential equation is homogeneous.

PuttingÂ

Â

Â

Putting these values ofÂ Â andÂ Â in eq. (i), we have

Â

Â

Â

Â

Integrating both sides,

Â

Â

Â

Â

Â

Â

PuttingÂ

Â

Â Â

Â Â where C =Â Â Â â€¦â€¦â€¦.(ii)

Now puttingÂ Â andÂ Â in eq. (ii), we get 1 = 3CÂ Â

Putting value of C in eq. (ii),

Â

###
**13.Â ****
****Â whenÂ ****
**

**Ans.Â**Given: Differential equation

Â

Â Â =Â Â Â â€¦â€¦â€¦.(i)

Therefore, the given differential equation is homogeneous.

PuttingÂ

Â

Â

Putting these values ofÂ Â andÂ Â in eq. (i), we have

Â

Â Â [Separating variables]

Integrating both sides,

Â

Â

Â Â [PuttingÂ ] â€¦â€¦â€¦.(ii)

Now puttingÂ Â in eq. (ii),

Â

Putting the value ofÂ Â in eq. (ii),

Â

Â

###
**14.Â ****
****Â whenÂ ****
**

**Ans.Â**Given: Differential equation

Â Â Â â€¦â€¦â€¦(i)

Therefore, the given differential equation is homogeneous.

PuttingÂ

Â

Â

Putting these values ofÂ Â andÂ Â in eq. (i), we have

Â

Â

Â Â [Separating variables]

Integrating both sides,

Â

Â

Â Â [PuttingÂ ] â€¦â€¦â€¦.(ii)

Now puttingÂ Â in eq. (ii),

Â

Putting the value ofÂ Â in eq. (ii),

Â

Â

###
**15.Â ****
****Â whenÂ ****
**

**Ans.Â**Given: Differential equation Â â€¦â€¦â€¦.(i)

Â

Â

Â Â â€¦â€¦â€¦.(ii)

Therefore the given differential equation is homogeneous because each coefficient ofÂ Â andÂ Â is same.

PuttingÂ

Â

Â

Putting these values ofÂ Â andÂ Â in eq. (ii), we have

Â

Â

Â Â [Separating variables]

Integrating both sides,

Â

Â

Â Â [PuttingÂ ]

Now puttingÂ Â inÂ ,

Â

Again puttingÂ , inÂ , we get

Â

Â

Â

**Choose the correct answer:**

**16. A homogeneous differential equation of the formÂ ****
****Â can be solved by making the substitution:**

**(A)Â
Â **

**(B)Â
Â **

**(C)Â
Â **

**(D)Â
Â **

**Ans.Â**We know that a homogeneous differential equation of the formÂ Â can be solved by the substitutionÂ Â i.e.,Â

Therefore, option (C) is correct.

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

**(A)Â ****
****Â **

**(B)Â ****
****Â **

**(C)Â ****
****Â **

**(D)Â ****
****Â **

**Ans.Â**D