Hello Students. In this post, you will find the complete** ****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.**

(*x*² + *xy*) *dy* = (*x*² + *y*²) *dx*

**Solution:**

The given differential equation is (*x*² + *xy*) *dy* = (*x*² + *y*²) *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

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

**Maths Class 12 Ex 9.4 Question 3.**

(*x* – *y*) *dy* – (*x* + *y*) *dx* = 0

**Solution:**

The given differential equation is (*x* – *y*) *dy* – (*x* + *y*) *dx* = 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

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

**Maths Class 12 Ex 9.4 Question 4.**

(*x*² – *y*²) *dx* + 2*xy dy* = 0

**Solution:**

The given differential equation is (*x*² – *y*²) *dx* + 2*xy dy* = 0

*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

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

*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

*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

*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

*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

*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

*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* + *y*) *dy* + (*x* – *y*) *dx* = 0, *y* = 1 when *x* = 1

**Solution:**

The given differential equation is (*x* + *y*) *dy* + (*x* – *y*) *dx* = 0, *y* = 1 when *x* = 1

*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

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.**

*x*² *dy* + (*xy* + *y*²) *dx* = 0, *y* = 1 when *x* = 1

**Solution:**

The given differential equation is *x*² *dy* + (*xy* + *y*²) *dx* = 0, *y* = 1 when *x* = 1

*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

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

*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

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

It is given that *y* = Ï€/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

*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

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

*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

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) (4*x* + 6*y* + 5) *dy* – (3*y* + 2*x* + 4) *dx* = 0

(B) (*xy*) *dx* – (*x*^{3} + *y*^{3}) *dy* = 0

(C) (*x*^{3} + 2*y*^{2}) *dx* + 2*xy dy* = 0

(D) *y*^{2} *dx* + (*x*^{2} – *xy* – *y*^{2}) *dy* = 0

**Solution:**

(D)

Here, *f*(*x*, *y*) 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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