Class 12 NCERT Solutions- Mathematics Part ii – Chapter 9– Differential Equations Exercise 9.5

In the article, we will solve Exercise 9.5 from Chapter 9, “Differential Equations” in the NCERT. Exercise 9.5 covers Homogeneous differential equations.

Basic Concept to Solve Exercise 9.5

STEP 1: First of all, prove that the given differential equation is a Homogeneous differential equation.

STEP 2: Then put in the differential equation.

STEP 3: Then solve by using the variable separation method.

Exercise 9.5 Solution

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

Q.1:

Solution:

We have This equation can be written as -

Let

Thus, Equation is a homogeneous equation.

Let

Differentiating both sides with respect to x we get:

Integrating both sides:

Hence the required solution is

Q.2:

Solution:

can be written as

Let

Thus, Equation is a homogeneous equation.

Let

Differentiating both sides we get:

On Integrating:

Hence the required solution is

Q.3:

Solution:

can be written as

Thus, Equation is a homogeneous equation.

Let

On Differentiating:

From above, we have

, Substituting these values:

So,

Q.4:

Solution:

can be written as

Let

Thus, differential equation is a homogeneous equation.

Let

Q.5:

Solution:

can be written as

Thus, given differential equation is a homogeneous equation.

Q.6:

Solution:

cab be written as

Let

Thus, Given differential equation is a homogeneous equation.

Let

Q.7:

Solution:

can be written as

Thus, Given differential equation is a homogeneous equation.

Let

Q.8:

Solution:

can be written as

Thus, Given differential equation is a homogeneous equation.

Let

Q.9:

Solution:

can be written as

Let

Thus, Given differential equation is a homogeneous equation

Let

Let

So,

Q.10:

Solution:

can be written as

Thus, Given differential equation is a homogeneous equation.

Let

For each of the differential equations in Exercises from 11 to 15, find the particular solution satisfying the given condition.

Q.11: when

Solution:

can be written as

Thus, Given differential equation is a homogeneous equation.

Let

Now, put

we get,

Q.12: when

Solution:

can be written as

Thus, Given differential equation is a homogeneous equation.

Let

Now, put y = 1 and x = 1

Q.13:

Solution:

can be written as

Thus, Given differential equation is a homogeneous equation.

Let

Now, put

Q.14:

Solution:

can be written as

Thus, Given differential equation is a homogeneous equation.

Let

Q.15:

Solution:

can be written as

Let

Thus, Given differential equation is a homogeneous equation.

Let

Q.16: A homogeneous differential equation of the form can be solved by making the substitution.

(A)

(B)

(C)

(D)

Solution:

For solving homogeneous equation of form , we need to make substitution as

.Thus, the correct option is C.

Q.17: Which of the following is a homogeneous differential equation?

(A)

(B)

(C)

(D)

Solution:

is homogeneous function of degree n , if for non-zero constant

.Consider equation given in D

which can be written as

Thus, Differential equation given in D is a homogeneous equation.

Also Check,

• Chapter 9 Differential Equations-Exercise -9.1
• Chapter 9 Differential Equations-Exercise -9.2
• Chapter 9 Differential Equations-Exercise -9.3