# NCERT solution class 12 chapter 3 Differential Equations exercise 3.5 mathematics part 2

## EXERCISE 3.5

#### Question 1:

The given differential equation i.e., (x2 + xydy = (x2 + y2dx can be written as:

This shows that equation (1) is a homogeneous equation.

To solve it, we make the substitution as:

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:

The given differential equation is:

Thus, the given equation is a homogeneous equation.

To solve it, we make the substitution as:

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:

The given differential equation is:

Thus, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

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:

The given differential equation is:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

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:

The given differential equation is:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

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:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

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:

The given differential equation is:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

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:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

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:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

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:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

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:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

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 2k in equation (2), we get:

This is the required solution of the given differential equation.

#### Question 12:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

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:

Therefore, the given differential equation is a homogeneous equation.

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

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:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

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:

Therefore, the given differential equation is a homogeneous equation.

To solve it, we make the substitution as:

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. vy

D. x = v

For solving the homogeneous equation of the form, we need to make the substitution as x = vy.

Hence, the correct answer is C.

#### Question 17:

Which of the following is a homogeneous differential equation?

A.

B.

C.

D.