NCERT Solutions for Maths Class 12 Exercise 9.4

NCERT Solutions for Maths Class 12 Exercise 9.4

Hello Students. In this post, you will find the complete NCERT Solutions for Maths Class 12 Exercise 9.4.

You can download the PDF of NCERT Books Maths Chapter 9 for your easy reference while studying NCERT Solutions for Maths Class 12 Exercise 9.4.

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NCERT Solutions for Maths Class 12 Exercise 9.1

NCERT Solutions for Maths Class 12 Exercise 9.2

NCERT Solutions for Maths Class 12 Exercise 9.3

NCERT Solutions for Maths Class 12 Exercise 9.5


NCERT Solutions for Maths Class 12 Exercise 9.4

 

In each of the Questions 1 to 10, show that the given differential equation is homogeneous and solve each of them.

Maths Class 12 Ex 9.4 Question 1.

(x² + xydy = (x² + y²) dx

Solution:

The given differential equation is (x² + xydy = (x² + y²) dx

 

From equation (i), we can see that dy/dx is in the form of f(y/x), so it is a homogeneous differential equation.

To solve it, put y = vx          … (ii)

Differentiating equation (ii) w.r.t. x, we get

Where C is an arbitrary constant.

This is the required general solution of the given differential equation.

Maths Class 12 Ex 9.4 Question 2.


Solution:
The given differential equation is 

Or dy/dx = 1 + y/x      … (i)

From equation (i), we can see that dy/dx is in the form of f(y/x), so it is a homogeneous differential equation.

To solve it, put y = vx          … (ii)

Differentiating equation (ii) w.r.t. x, we get

Where C is an arbitrary constant.

This is the required general solution of the given differential equation.

Maths Class 12 Ex 9.4 Question 3.

(x – ydy – (x + ydx = 0

Solution:

The given differential equation is (x – ydy – (x + ydx = 0

Or         … (i)

From equation (i), we can see that dy/dx is in the form of f(y/x), so it is a homogeneous differential equation.

To solve it, put y = vx          … (ii)

Differentiating equation (ii) w.r.t. x, we get

Where C is an arbitrary constant.

This is the required general solution of the given differential equation.

Maths Class 12 Ex 9.4 Question 4.

(x² – y²) dx + 2xy dy = 0

Solution:

The given differential equation is (x² – y²) dx + 2xy dy = 0

 

From equation (i), we can see that dy/dx is in the form of f(y/x), so it is a homogeneous differential equation.

To solve it, put y = vx          … (ii)

Differentiating equation (ii) w.r.t. x, we get

Where C is an arbitrary constant.

This is the required general solution of the given differential equation.

Maths Class 12 Ex 9.4 Question 5.

Solution:

The given differential equation is

 

From equation (i), we can see that dy/dx is in the form of f(y/x), so it is a homogeneous differential equation.

To solve it, put y = vx          … (ii)

Differentiating equation (ii) w.r.t. x, we get

Where C is an arbitrary constant.

This is the required general solution of the given differential equation.

Maths Class 12 Ex 9.4 Question 6.

Solution:

The given differential equation is

 

From equation (i), we can see that dy/dx is in the form of f(y/x), so it is a homogeneous differential equation.

To solve it, put y = vx          … (ii)

Differentiating equation (ii) w.r.t. x, we get

Where C is an arbitrary constant.

This is the required general solution of the given differential equation.

Maths Class 12 Ex 9.4 Question 7.

Solution:

The given differential equation is

 

From equation (i), we can see that dy/dx is in the form of f(y/x), so it is a homogeneous differential equation.

To solve it, put y = vx          … (ii)

Differentiating equation (ii) w.r.t. x, we get

Where C is an arbitrary constant.

This is the required general solution of the given differential equation.

Maths Class 12 Ex 9.4 Question 8.

Solution:

The given differential equation is

 

From equation (i), we can see that dy/dx is in the form of f(y/x), so it is a homogeneous differential equation.

To solve it, put y = vx          … (ii)

Differentiating equation (ii) w.r.t. x, we get

Where C is an arbitrary constant.

This is the required general solution of the given differential equation.

Maths Class 12 Ex 9.4 Question 9.

Solution:

The given differential equation is

 

From equation (i), we can see that dy/dx is in the form of f(y/x), so it is a homogeneous differential equation.

To solve it, put y = vx          … (ii)

Differentiating equation (ii) w.r.t. x, we get

Where C is an arbitrary constant.

This is the required general solution of the given differential equation.

Maths Class 12 Ex 9.4 Question 10.

Solution:

The given differential equation is

 

From equation (i), we can see that dy/dx is in the form of f(y/x), so it is a homogeneous differential equation.

To solve it, put y = vx          … (ii)

Differentiating equation (ii) w.r.t. x, we get

Where C is an arbitrary constant.

This is the required general solution of the given differential equation.

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

Maths Class 12 Ex 9.4 Question 11.

(x + ydy + (x – ydx = 0, y = 1 when x = 1

Solution:

The given differential equation is (x + ydy + (x – ydx = 0, y = 1 when x = 1

 

From equation (i), we can see that dy/dx is in the form of f(y/x), so it is a homogeneous differential equation.

To solve it, put y = vx          … (ii)

Differentiating equation (ii) w.r.t. x, we get

Where C is an arbitrary constant.

This is the required general solution of the given differential equation.

It is given that y = 1 when x = 1.

Putting these values of x and y in equation (iii), we get

This is the required particular solution of the given differential equation.

Maths Class 12 Ex 9.4 Question 12.

x² dy + (xy + y²) dx = 0, y = 1 when x = 1

Solution:

The given differential equation is x² dy + (xy + y²) dx = 0, y = 1 when x = 1

 

From equation (i), we can see that dy/dx is in the form of f(y/x), so it is a homogeneous differential equation.

To solve it, put y = vx          … (ii)

Differentiating equation (ii) w.r.t. x, we get

Where C is an arbitrary constant.

This is the required general solution of the given differential equation.

Maths Class 12 Ex 9.4 Question 13.

Solution:

The given differential equation is

 

From equation (i), we can see that dy/dx is in the form of f(y/x), so it is a homogeneous differential equation.

To solve it, put y = vx          … (ii)

Differentiating equation (ii) w.r.t. x, we get

Where C is an arbitrary constant.

This is the required general solution of the given differential equation.

It is given that y = π/4 when x = 1.

Putting these values of x and y in equation (iii), we get

This is the required particular solution of the given differential equation.

Maths Class 12 Ex 9.4 Question 14.

Solution:

The given differential equation is

      … (i)

From equation (i), we can see that dy/dx is in the form of f(y/x), so it is a homogeneous differential equation.

To solve it, put y = vx          … (ii)

Differentiating equation (ii) w.r.t. x, we get

Where C is an arbitrary constant.

This is the required general solution of the given differential equation.

It is given that y = 0 when x = 1.

Putting these values of x and y in equation (iii), we get

This is the required particular solution of the given differential equation.

Maths Class 12 Ex 9.4 Question 15.

Solution:

The given differential equation is

From equation (i), we can see that dy/dx is in the form of f(y/x), so it is a homogeneous differential equation.

To solve it, put y = vx          … (ii)

Differentiating equation (ii) w.r.t. x, we get

Where C is an arbitrary constant.

This is the required general solution of the given differential equation.

It is given that y = 2 when x = 1.

Putting these values of x and y in equation (iii), we get

This is the required particular solution of the given differential equation.

Maths Class 12 Ex 9.5 Question 16.

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

(A) y = vx
(B) v = yx
(C) x = vy
(D) x = v

Solution:

(C) We know that the homogeneous equation of the form dx/dy = h(x/y) can be solved by putting x = vy. Hence, the correct answer is option (C).


Maths Class 12 Ex 9.4 Question 17.

Which of the following is a homogeneous differential equation?
(A) (4x + 6y + 5) dy – (3y + 2x + 4) dx = 0
(B) (xydx – (x3 + y3dy = 0
(C) (x3 + 2y2dx + 2xy dy  = 0
(D) y2 dx + (x2 – xy – y2dy = 0

Solution:
(D)

Here, f(xy) is a homogeneous function of degree zero. Therefore, it is a homogeneous differential equation.

Hence, the correct answer is option (D).



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