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Question 1:

Show that the given differential equation is homogeneous and solve them:

Solution 1:

The given differential equation i.e., (x² + xy) dy = (x² + y²) 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.

Comments

Nurullmobeen

Question 2:

Show that the given differential equation is homogeneous and solve them:

Solution 2:

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.

Comments

Nurullmobeen

Question 3:

Show that the given differential equation is homogeneous and solve them:

Solution 3:

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.

Comments

Nurullmobeen

Question 4:

Show that the given differential equation is homogeneous and solve them:

Solution 4:

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:

Comments

Nurullmobeen

Question 5:

Show that the given differential equation is homogeneous and solve them:

Solution 5:

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.

Comments

Nurullmobeen

Question 6:

Show that the given differential equation is homogeneous and solve them:

Solution 6:

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.

Comments

Nurullmobeen

Question 7:

Show that the given differential equation is homogeneous and solve them:

Solution 7:

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.

Comments

Nurullmobeen

Question 8:

Show that the given differential equation is homogeneous and solve them:

Solution 8:

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.

Comments

Tasneem

Question 9:

Show that the given differential equation is homogeneous and solve them:

Solution 9:

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.

Comments

Nurullmobeen

Question 10:

Show that the given differential equation is homogeneous and solve them:

Solution 10:

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.

Comments

Nurullmobeen

Question 11:

For the given differential equation , find the particular solution satisfying the given condition:

Solution 11:

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 2k in equation (2), we get:

This is the required solution of the given differential equation.

Comments

Nurullmobeen

Question 12:

For the given differential equation , find the particular solution satisfying the given condition:

Solution 12:

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.

Comments

Nurullmobeen

Question 13:

For the given differential equation , find the particular solution satisfying the given condition:

Solution 13:

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.

Comments

Nurullmobeen

Question 14:

For the given differential equation , find the particular solution satisfying the given condition:

Solution 14:

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.

Comments

Nurullmobeen

Question 15:

For the given differential equation , find the particular solution satisfying the given condition:

Solution 15:

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.

Comments

Nurullmobeen

Question 16:

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

Solution 16:

For solving the homogeneous equation of the form, we need to make the substitution as x = vy.

Hence, the correct answer is C.

Comments

Nurullmobeen

Question 17:

Which of the following is a homogeneous differential equation?

Solution 17:

Function F(x, y) is said to be the homogenous function of degree n, if

F(λx, λy) = λⁿ 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.

Comments

Tasneem

All Exercises - Chapter 9 - Differential Equations

Exercise 9.5

Miscellaneous Exercise 9

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Learn Exercise 9.5 with Free Lessons & Tips

All Exercises - Chapter 9 - Differential Equations

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