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

#### 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.

Function F(xy) is said to be the homogenous function of degree n, if
F(λx, λy) = λn F(xy) for any non-zero constant (λ).
Consider the equation given in alternativeD:
Hence, the differential equation given in alternative D is a homogenous equation.

Courtesy : CBSE