## EXERCISE 3.5

#### 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.