## NCERT Solutions for Class 12 Maths Chapter 9 – Differential Equations Ex 9.5

#### Page No 406:

#### Question 1:

#### Answer:

The given differential equation i.e., (

*x*^{2}+*xy*)*dy*= (*x*^{2}+*y*^{2})*dx*can be written as:
This shows that equation (1) is a homogeneous equation.

To solve it, we make the substitution as:

*y*=

*vx*

Differentiating both sides with respect to

*x*, we get:
Substituting the values of

*v*and in equation (1), we get:
Integrating both sides, we get:

This is the required solution of the given differential equation.

#### Question 2:

#### Answer:

The given differential equation is:

Thus, the given equation is a homogeneous equation.

To solve it, we make the substitution as:

*y*=

*vx*

Differentiating both sides with respect to

*x*, we get:
Substituting the values of

*y*and in equation (1), we get:
Integrating both sides, we get:

This is the required solution of the given differential equation.

#### Question 3:

#### Answer:

The given differential equation is:

Thus, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

*y*=

*vx*

Substituting the values of

*y*and in equation (1), we get:
Integrating both sides, we get:

This is the required solution of the given differential equation.

#### Question 4:

#### Answer:

The given differential equation is:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

*y*=

*vx*

Substituting the values of

*y*and in equation (1), we get:
Integrating both sides, we get:

This is the required solution of the given differential equation.

#### Question 5:

#### Answer:

The given differential equation is:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

*y*=

*vx*

Substituting the values of

*y*and in equation (1), we get:
Integrating both sides, we get:

This is the required solution for the given differential equation.

#### Question 6:

#### Answer:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

*y*=

*vx*

Substituting the values of

*v*and in equation (1), we get:
Integrating both sides, we get:

This is the required solution of the given differential equation.

#### Question 7:

#### Answer:

The given differential equation is:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

*y*=

*vx*

Substituting the values of

*y*and in equation (1), we get:
Integrating both sides, we get:

This is the required solution of the given differential equation.

#### Question 8:

#### Answer:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

*y*=

*vx*

Substituting the values of

*y*and in equation (1), we get:
Integrating both sides, we get:

This is the required solution of the given differential equation.

#### Question 9:

#### Answer:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

*y*=

*vx*

Substituting the values of

*y*and in equation (1), we get:
Integrating both sides, we get:

Therefore, equation (1) becomes:

This is the required solution of the given differential equation.

#### Question 10:

#### Answer:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

*x*=

*vy*

Substituting the values of

*x*and in equation (1), we get:
Integrating both sides, we get:

This is the required solution of the given differential equation.

#### Question 11:

#### Answer:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

*y*=

*vx*

Substituting the values of

*y*and in equation (1), we get:
Integrating both sides, we get:

Now,

*y*= 1 at*x*= 1.
Substituting the value of 2

*k*in equation (2), we get:
This is the required solution of the given differential equation.

#### Question 12:

#### Answer:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

*y*=

*vx*

Substituting the values of

*y*and in equation (1), we get:
Integrating both sides, we get:

Now,

*y*= 1 at*x*= 1.
Substituting in equation (2), we get:

This is the required solution of the given differential equation.

#### Question 13:

#### Answer:

Therefore, the given differential equation is a homogeneous equation.

To solve this differential equation, we make the substitution as:

*y*=

*vx*

Substituting the values of

*y*and in equation (1), we get:
Integrating both sides, we get:

Now, .

Substituting C =

*e*in equation (2), we get:
This is the required solution of the given differential equation.

#### Question 14:

#### Answer:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

*y*=

*vx*

Substituting the values of

*y*and in equation (1), we get:
Integrating both sides, we get:

This is the required solution of the given differential equation.

Now,

*y*= 0 at*x*= 1.
Substituting C =

*e*in equation (2), we get:
This is the required solution of the given differential equation.

#### Question 15:

#### Answer:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

*y*=

*vx*

Substituting the value of

*y*and in equation (1), we get:
Integrating both sides, we get:

Now,

*y*= 2 at*x*= 1.
Substituting

*C*= –1 in equation (2), we get:
This is the required solution of the given differential equation.

#### Question 16:

A homogeneous differential equation of the form can be solved by making the substitution

**A.**

*y*=

*vx*

**B.**

*v*=

*yx*

**C.**

*x*=

*vy*

**D.**

*x*=

*v*

#### Answer:

For solving the homogeneous equation of the form, we need to make the substitution as

*x*=*vy*.
Hence, the correct answer is C.

#### Page No 407:

#### Question 17:

Which of the following is a homogeneous differential equation?

**A.**

**B.**

**C.**

**D.**

#### Answer:

Function F(

*x*,*y*) is said to be the homogenous function of degree*n,*if
F(λ

*x*, λ*y*) = λ^{n}F(*x*,*y*) for any non-zero constant (λ).
Consider the equation given in alternativeD:

Hence, the differential equation given in alternative

**D**is a homogenous equation._{}

^{}