Class 12 RD Sharma Solutions- Chapter 22 Differential Equations - Exercise 22.1 | Set 1

Determine the order and degree of the following differential equation. State also whether it is linear or non-linear(Questions 1-13)

Question 1. 

Solution:

We have,

Order of function:

The Highest order of derivative of function is 3 i.e.,

So, the order of derivative is equal to 3.

Degree of function:

As the power of the highest order derivative of function is 1 (i.e., power of  is 1)

So, degree of function is 1.

Linear or Non-linear:

The given equation is non-linear.

Question 2. 

Solution:

We have,

Order of function:

As the highest order of derivative of function is 2.(i.e.,)

So, Order of the function is equal to 2.

Degree of function:

As the power of the highest order derivative of the function is 1(i.e., power of  is 1)

So, Degree of the function is equal to 1.

Linear or Non-linear:

The given equation is linear.

Question 3. 

Solution:

We have,

Order of function:

As the highest order of derivative of function is 1 (i.e., )

So, Order of the function is equal to 1.

Degree of function

As the power of the highest order derivative of the function is 3 (i.e., power of dy/dx is 3)

So, the degree of the function is equal to 3.

Linear or Non-linear:

The given equation is non-linear.

Question 4. 

Solution:

We have,

On squaring both side, we get

On cubing both side, we get

Order of function:

As the highest order of derivative of function is 2 (i.e.,

So, Order of the function is equal to 2.

Degree of function:

As the power of the highest order derivative of the function is 2. (i.e., power of  is 2)

So, the Degree of the function is equal to 2.

Linear or Non-linear:

The given equation is non-linear.

Question 5. 

Solution:

We have,

Order of function:

As the highest order of derivative of function is 2 

So, Order of the function is equal to 2.

Degree of function:

As the power of the highest order derivative of function is 1 (i.e., power of  is 1)

So, the Degree of the function is equal to 1.

Linear or Non-linear:

The given equation is non-linear.

Question 6. 

Solution:  

We have,

On cubing both side, we get

On squaring both side, we get

Order of function:

As the highest order of derivative of function is 2 (i.e., )

So, the Order of the function is equal to 2.

Degree of function:

As the power of the highest order derivative of the function is 2(i.e., power of  is 2)

So, the Degree of the function is equal to 2.

Linear or Non-linear:

The given equation is non-linear.

Question 7. 

Solution:

We have,

On squaring both side, we get

Order of function:

The highest order of derivative of function is 4 (i.e., )

So, the order of the derivative is equal to 4.

Degree of function:

As the power of the highest order derivative of the function is 2 (i.e., power of  is 2)

So, the degree of function is 2.

Linear or Non-linear:

The given equation is non-linear.

Question 8: 

Solution:

We have,

On squaring both side, we have

Order of function:

As the highest order of derivative of function is 1. 

So, the Order of the function is equal to 1.

Degree of function:

As the power of the highest order derivative of the function is 1.

So, the degree of the function is equal to 1.

Linear or Non-linear:

The given equation is linear.

Question 9: 

Solution:

We have,

Order of function:

As the highest order of derivative of function is 2 (i.e.,)

So, order of derivative is equal to 2.

Degree of function:

As the power of the highest order derivative of the function is 1 (i.e., power of is 1)

So, the Degree of the function is equal to 1.

Linear or Non-linear:

The given equation is linear.

Question 10: 

Solution:

We have,

Order of function:

As the highest order of derivative of the function is 2.

So, the Order of the function is equal to 2.

Degree of function:

As the power of the highest order derivative of the function is 1 (i.e., power of  is 1)

So, the Degree of the function is equal to 1.

Linear or Non-linear:

The given equation is non-linear.

Question 11: 

Solution:

We have,

Order of function:

As the highest order of derivative of the function is 2

So, the Order of the function is equal to 2.

Degree of function:

As the power of the highest order derivative of the function is 3. (i.e., power of is 3)

So, the degree of the function is equal to 3.

Linear or Non-linear:

The given equation is non-linear.

Question 12: 

Solution:

We have,

Order of function:

As the highest order of derivative of the function is 3

So, the Order of the function is equal to 3.

Degree of function:

As the power of the highest order derivative of the function is 1.(i.e., power of is 1)

So, the Degree of the function is equal to 1.

Linear or Non-linear:

The given equation is non-linear.

Question 13: 

Solution:

We have,

Order of function:

As the highest order of derivative of the function is 1

So, the Order of the function is equal to 1.

Degree of function:

As the power of the highest order derivative of the function is 1. (i.e., power of dy/dx is 1)

So, the Order of the function is equal to 1.

Linear or Non-linear:

The given equation is non-linear.

Practice Questions

1. Verify that y = e^(x^2) is a solution of the differential equation dy/dx = 2xy.

2. Find the order and degree of the differential equation: (d^2y/dx^2 + 1)^3 = 0

3. Form the differential equation of the family of curves y = ax^2 + bx + c, where a, b, and c are arbitrary constants.

4. Solve the differential equation: dy/dx + y tan x = sec x

5. Find the general solution of the differential equation: dy/dx = x/y

6. Verify that y = sin x + cos x is a solution of the differential equation d^2y/dx^2 + y = 0

7. Solve the differential equation: (1 + x^2)dy/dx = 1 - y^2

8. Find the particular solution of dy/dx = 2x, given that y = 1 when x = 0.

9. Form the differential equation of all circles with center at the origin.

10. Solve the differential equation: dy/dx = (y^2 - 1)/(2xy)

Summary

Chapter 22 of RD Sharma's Class 12 mathematics textbook focuses on Differential Equations.

Questions

• Definition of a differential equation
• Order and degree of a differential equation
• Formation of differential equations by eliminating arbitrary constants
• Solutions of first-order differential equations
• Methods for solving separable variables
• Homogeneous differential equations
• Linear differential equations
• Exact differential equations

This chapter provides a foundation for understanding and solving various types of differential equations, which are crucial in many areas of mathematics, physics, and engineering.