##### Tallentex 2025 - ALLEN's Talent Encouragement Exam

ApplyRegister for Tallentex '25 - One of The Biggest Talent Encouragement Exam

Edited By Satyajeet Kumar | Updated on Jan 24, 2022 04:19 PM IST

Class 12 students face regular tests and exams to test their understanding ability of almost every concept they learn. This requires regular practice in the right manner. Topics like Differential equations in mathematics are always challenging for high school students. Students must have the RD Sharma Class 12th Exercise 21.2 to have clarity in this chapter and attend those sums without any doubts. Not only do students use these solution books, many teachers and tutors also do to recheck their answers and find easier methods to solve the sums.

Differential Equation exercise 21.2 question 1

Answer:

Hint:

Differentiating the given equation

Given:

Solution:

Differential Equation exercise 21.2 question 2

Answer:

Hint:

Differentiating given equation with respect to x

[Applying log both sides]

Given:

Solution:

We have,

Hence,

is the differential equation corresponding to

Hint:

Differentiating given equation with respect to x

[Applying log both sides]

Given:

Solution:

We have,

Hence,

is the differential equation corresponding to

Differential Equation exercise 21.2 question 3 (i)

Answer:

Hint:

Differentiating the given equation with respect to x

Given:

Solution:

On differentiating with respect to x,

Substituting the value of a, we get

Hence,

is the differential equation corresponding to

Hint:

Differentiating the given equation with respect to x

Given:

Solution:

On differentiating with respect to x,

Substituting the value of a, we get

Hence,

is the differential equation corresponding to

Differential Equation exercise 21.2 question 3 (ii)

Answer:

Hint:

Differentiating the given curve, c- Parameter

Given:

Solution:

Hint:

Differentiating the given curve, c- Parameter

Given:

Solution:

Differential Equation exercise 21.2 question 3 (iii)

Answer:

Hint:

Differentiating the given equation, where a-Parameter

Given:

Solution:

Hint:

Differentiating the given equation, where a-Parameter

Given:

Solution:

Differential Equation exercise 21.2 question 3 (iv)

Answer:

Hint:

Differentiating the given equation

Given:

Solution:

Hint:

Differentiating the given equation

Given:

Solution:

Differential Equation exercise 21.2 question 4

Answer:

Hint:

Differentiating the given equation in first and second order

Given:

Solution:

Hint:

Differentiating the given equation in first and second order

Given:

Solution:

Differential Equation exercise 21.2 question 5

Answer:

Hint:

Differentiating the given equation with respect to x.

Let the same equation and substitutes its value.

Given:

Solution:

We get,

Hint:

Differentiating the given equation with respect to x.

Let the same equation and substitutes its value.

Given:

Solution:

We get,

Differential Equation exercise 21.2 question 6

Answer:

Hint:

Differentiating the given equation, let it equation (i) and substitute a from (iii) in (ii)

Given:

Solution:

Again,

Substitute equation (iii) in (ii),

Hint:

Differentiating the given equation, let it equation (i) and substitute a from (iii) in (ii)

Given:

Solution:

Again,

Substitute equation (iii) in (ii),

Differential Equation exercise 21.2 question 7

Answer:

Hint:

Differentiating the given equation with respect to x

Given:

Solution:

Differentiating with respect to x

Putting the value of a in the given equation, we get

put

Hint:

Differentiating the given equation with respect to x

Given:

Solution:

Differentiating with respect to x

Putting the value of a in the given equation, we get

put

Differential Equation exercise 21.2 question 8

Answer:

Hint:

Differentiating the given equation where r is a constant

Given:

Solution:

From equation (ii)

Put this value in equation (i)

Squaring equation (iii),

Dividing equation (iv) by (v),

Hint:

Differentiating the given equation where r is a constant

Given:

Solution:

From equation (ii)

Put this value in equation (i)

Squaring equation (iii),

Dividing equation (iv) by (v),

Differential Equation exercise 21.2 question 9

Answer:

Hint:

Let (0, k) be the centre of the circle with k as its centre and differentiating the equation of the circle

Given:

Circles pass through origin and their centre lie on y-axis

Solution:

Hint:

Let (0, k) be the centre of the circle with k as its centre and differentiating the equation of the circle

Given:

Circles pass through origin and their centre lie on y-axis

Solution:

Differential Equation exercise 21.2 question 10

Answer:Hint:

Let the equation of the circle assuming R as radius and (a,b) as the centre of the circle is

Given:

Circles passes through origin and their centre lie on x-axis. So the centre of the circle will be (a, 0)

Solution:

Substituting these values in equation (i)

Differential Equation exercise 21.2 question 11

Answer:

Hint:

Let the surface be A

Rate of evaporation,

Using formula,

Given:

Rain drop evaporates at a rate proportional to its surface area

Solution:

Hint:

Let the surface be A

Rate of evaporation,

Using formula,

Given:

Rain drop evaporates at a rate proportional to its surface area

Solution:

Differential Equation exercise 21.2 question 12

Answer:

Hint:

Let the equation of parabola be

Where h, k are parameter

Given:

Latus rectum 4a and axis are parallel to x-axis

Solution:

Let the equation of parabola be

Where h, k are parameter

which is the required differential equation.

Hint:

Let the equation of parabola be

Where h, k are parameter

Given:

Latus rectum 4a and axis are parallel to x-axis

Solution:

Let the equation of parabola be

Where h, k are parameter

which is the required differential equation.

Differential Equation exercise 21.2 question 14

Answer:

Hint:

Differentiating the given equation with respect to x

Given:

Solution:

Hint:

Differentiating the given equation with respect to x

Given:

Solution:

Differential Equation exercise 21.2 question 15 (i)

Answer:

Hint:

Using chain rule and putting the value of a in (i)

Given:

Solution:

Hint:

Using chain rule and putting the value of a in (i)

Given:

Solution:

Differential Equation exercise 21.2 question 15 (ii)

Answer:

Hint:

Differentiating the given equation

Given:

Solution:

Hint:

Differentiating the given equation

Given:

Solution:

Differential Equation exercise 21.2 question 15.3 (iii)

Answer:

Hint:

Using the formula (a+b)^{2} and differentiating the given equation

Given:

Solution:

Hint:

Using the formula (a+b)

Given:

Solution:

Differential Equation exercise 21.2 question 16 (i)

Answer:

The required equation is

Hint:

The given equation is the equation of a circle

Given:

Solution:

Differentiating with respect to x,

The required equation is

The required equation is

Hint:

The given equation is the equation of a circle

Given:

Solution:

Differentiating with respect to x,

The required equation is

Differential Equation exercise 21.2 question 16 (ii)

Answer:

The required equation is

Hint:

Differentiating the given equation with respect to x

Given:

Solution:

Differentiating with respect to x,

The required equation is

The required equation is

Hint:

Differentiating the given equation with respect to x

Given:

Solution:

Differentiating with respect to x,

The required equation is

Differential Equation exercise 21.2 question 16 (iii)

Answer:

The required equation is

Hint:

The equation is the equation of parabola

Given:

Solution:

The required equation is

The required equation is

Hint:

The equation is the equation of parabola

Given:

Solution:

The required equation is

Differential Equation exercise 21.2 question 16 (iv)

Answer:

The required differential equation is

Hint:

Given:

Solution:

The equation of family of curves is

Where b is a parameter

Differentiating equation (i) with respect to x, we get

The required equation is

The required differential equation is

Hint:

Given:

Solution:

The equation of family of curves is

Where b is a parameter

Differentiating equation (i) with respect to x, we get

The required equation is

Differential Equation exercise 21.2 question 16 (v)

Answer:

The required differential equation is

Hint:

Differentiating equation (i) with respect to x

Given:

Solution:

The equation of family of curves is

Where a is a parameter

Differentiating equation (i) with respect to x, we get

The required equation is

The required differential equation is

Hint:

Differentiating equation (i) with respect to x

Given:

Solution:

The equation of family of curves is

Where a is a parameter

Differentiating equation (i) with respect to x, we get

The required equation is

Differential Equation exercise 21.2 question 16 (vi)

Answer:

The required differential equation is

Hint:

This is the equation of hyperbola

Given:

Solution:

The equation of family of curves is

Where a and b are parameter

Differentiating equation (i) with respect to x, we get

Differentiating equation (ii) with respect to x, we get

Now, from (ii), we get

From (iii) and (iv), we get

The required differential equation is

The required differential equation is

Hint:

This is the equation of hyperbola

Given:

Solution:

The equation of family of curves is

Where a and b are parameter

Differentiating equation (i) with respect to x, we get

Differentiating equation (ii) with respect to x, we get

Now, from (ii), we get

From (iii) and (iv), we get

The required differential equation is

Differential Equation exercise 21.2 question 16 (vii)

Answer:

The required differential equation is

Hint:

Differentiating equation (i) and (ii) with respect to x

Given:

Solution:

The equation of family of curves is

Where a and b are parameters

Differentiating equation (i) with respect to x, we get

Differentiating equation (ii) with respect to x, we get

The required differential equation is

The required differential equation is

Hint:

Differentiating equation (i) and (ii) with respect to x

Given:

Solution:

The equation of family of curves is

Where a and b are parameters

Differentiating equation (i) with respect to x, we get

Differentiating equation (ii) with respect to x, we get

The required differential equation is

Differential Equation exercise 21.2 question 16 (viii)

Answer:

The required equation is

Hint:

Differentiating the given equation with respect to x

Given:

Solution:

On differentiating with respect to x, we get

From the given equation,

So, we have

The required equation is

The required equation is

Hint:

Differentiating the given equation with respect to x

Given:

Solution:

On differentiating with respect to x, we get

From the given equation,

So, we have

The required equation is

Differential Equation exercise 21.2 question 16 (ix)

Answer:

The required equation is

Hint:

Differentiating the given equation with respect to x

Given:

Solution:

Differentiating with respect to x,

The required equation is

The required equation is

Hint:

Differentiating the given equation with respect to x

Given:

Solution:

Differentiating with respect to x,

The required equation is

Differential Equation exercise 21.2 question 16 (x)

Answer:

The required equation is

Hint:

Differentiating the given equation with respect to x

Given:

Solution:

Differentiating with respect to x

From the given equation, we have

Now,

The required equation is

The required equation is

Hint:

Differentiating the given equation with respect to x

Given:

Solution:

Differentiating with respect to x

From the given equation, we have

Now,

The required equation is

Differential Equation exercise 21.2 question 17

Answer:

The required differential equation is

Hint:

Ellipse centre at the origin and foci on the x-axis

Given:

Ellipse having the centre at the origin and foci on the x-axis

Solution:

Equation of required ellipse is

Where a and b are arbitrary constants

Differentiating equation (i) with respect to x, we get

Differentiating equation (ii) with respect to x, we get

The required differential equation is

The required differential equation is

Hint:

Ellipse centre at the origin and foci on the x-axis

Given:

Ellipse having the centre at the origin and foci on the x-axis

Solution:

Equation of required ellipse is

Where a and b are arbitrary constants

Differentiating equation (i) with respect to x, we get

Differentiating equation (ii) with respect to x, we get

The required differential equation is

Differential Equation exercise 21.2 question 18

Answer:

The required differential equation is

Hint:

Hyperbola centre at the origin and foci on the x-axis

Given:

Hyperbolas having the centre at the origin and foci on the x-axis

Solution:

Equation of required ellipse is

Where a and b are arbitrary constants

Differentiating equation (i) with respect to x, we get

Differentiating equation (ii) with respect to x, we get

Substituting this value of 1/a^{2} in equation (ii), we get

The required differential equation is

The required differential equation is

Hint:

Hyperbola centre at the origin and foci on the x-axis

Given:

Hyperbolas having the centre at the origin and foci on the x-axis

Solution:

Equation of required ellipse is

Where a and b are arbitrary constants

Differentiating equation (i) with respect to x, we get

Differentiating equation (ii) with respect to x, we get

Substituting this value of 1/a

The required differential equation is

Differential Equation exercise 21.2 question 19

Answer:

The required differential equation is

Hint:

The equation belongs to the family of a circle

Given:

The family of circles in the second quadrant and touching the co-ordinate axes

Solution:

Let c denotes the family of circles in the second quadrant and touching the coordinate axes and let(-a,a) be coordinate of the centre of any member of this circle

Now, the equation representing this family of circle is

Differentiating (ii) with respect to x, we get

Substituting this value of a in (i), we get

The required differential equation is

The required differential equation is

Hint:

The equation belongs to the family of a circle

Given:

The family of circles in the second quadrant and touching the co-ordinate axes

Solution:

Let c denotes the family of circles in the second quadrant and touching the coordinate axes and let(-a,a) be coordinate of the centre of any member of this circle

Now, the equation representing this family of circle is

Differentiating (ii) with respect to x, we get

Substituting this value of a in (i), we get

The required differential equation is

Differential Equation exercise 21.2 question 20

Answer:

The required differential equation is

Hint:

Differentiating the given equation with respect to x

Given:

Solution:

Where a and b are arbitrary constants

Differentiating with respect to x

Again, Differentiating with respect to x

Put

from (i) and (ii) in equation (iii), we get

The required differential equation is

The required differential equation is

Hint:

Differentiating the given equation with respect to x

Given:

Solution:

Where a and b are arbitrary constants

Differentiating with respect to x

Again, Differentiating with respect to x

Put

from (i) and (ii) in equation (iii), we get

The required differential equation is

Differential Equation exercise 21.2 question 21

Answer:

The required differential equation is

Hint:

Differentiating the given equation with respect to x

Given:

Solution:

Differentiating with respect to x, we get

Putting (i) in (ii)

Again, Differentiating with respect to x, we get

Putting (iii) in (iv), we get

The required differential equation is

The required differential equation is

Hint:

Differentiating the given equation with respect to x

Given:

Solution:

Differentiating with respect to x, we get

Putting (i) in (ii)

Again, Differentiating with respect to x, we get

Putting (iii) in (iv), we get

The required differential equation is

Chapter 21, Differential Equation of class 12, is a portion where most of the students lose their marks. This chapter consists of eleven exercises, ex 21.1 to ex 21.11. this seconds exercise, ex 21.2, contains 27 sums to be solved. The concepts involved in this exercise are solving the differential equations in various forms. Even though the entire chapter has the same concept, the method by which each sum is solved varies greatly. Therefore, the use of RD Sharma Class 12 Chapter 21 Exercise 21.2 becomes essential to make the students understand each method without being confused by one another.

There is no other option than to practice the whole set of sums multiple times. The top-rated reference book, the RD Sharma Class 12th Exercise 21.2, comes with numerous practice questions for the students to work out. The more sums they practice in differential equations, the better they understand the concept in-depth. With the latest questions updated according to the NCERT syllabus, the Class 12 RD Sharma Chapter 21 Exercise 21.2 Solution is the best reference guide. For the sums that have various possible methods given in this book, students can select the ones that they feel to adapt.

Many teachers use the RD Sharma Class 12 Solutions Differential Equation Ex 21.2 reference material to frame their question papers. When the students practice with this book beforehand, they eventually face the questions easily during their tests and exams. RD Sharma solutions As the most suggested solution book by the previous bath students, the RD Sharma Class 12th Exercise 21.2 is popular among the class 12 students.

The best part of owning an RD Sharma book is that it is available for free of cost at the Career 360 website. Anyone can download the RD Sharma Class 12 Solutions Chapter 21 Ex 21.2 from the Career 360 website directly without wanting to pay even a penny. Many students who use these books for their exam preparation have admitted that their test scores have improved. Now, it is your turn to prepare for our grade 12 public exam with the RD Sharma books.

- Chapter 1 - Relations
- Chapter 2 - Functions
- Chapter 3 - Inverse Trigonometric Functions
- Chapter 4 - Algebra of Matrices
- Chapter 5 - Determinants
- Chapter 6 - Adjoint and Inverse of a Matrix
- Chapter 7 - Solution of Simultaneous Linear Equations
- Chapter 8 - Continuity
- Chapter 9 - Differentiability
- Chapter 10 - Differentiation
- Chapter 11 - Higher Order Derivatives
- Chapter 12 - Derivative as a Rate Measurer
- Chapter 13 - Differentials, Errors and Approximations
- Chapter 14 - Mean Value Theorems
- Chapter 15 - Tangents and Normals
- Chapter 16 - Increasing and Decreasing Functions
- Chapter 17 - Maxima and Minima
- Chapter 18 - Indefinite Integrals
- Chapter 19 - Definite Integrals
- Chapter 20 - Areas of Bounded Regions
- Chapter 21 - Differential Equations
- Chapter 22 - Algebra of Vectors
- Chapter 23 - Scalar Or Dot Product
- Chapter 24 - Vector or Cross Product
- Chapter 25 - Scalar Triple Product
- Chapter 26 - Direction Cosines and Direction Ratios
- Chapter 27 - Straight Line in Space
- Chapter 28 - The Plane
- Chapter 29 - Linear programming
- Chapter 30- Probability
- Chapter 31 - Mean and Variance of a Random Variable

1. Which reference books are very much helpful for the students who struggle with the chapter Differential Equation?

The RD Sharma Class 12th Exercise 21.2 reference book is the best guide for the students who are unclear about the sums in the Differential Equation chapter.

2. Where can the students find the RD Sharma books that can be accessed without payment?

Everyone can access the RD Sharma books at the Career 360 website without any payment.

3. How can a student improve their ability to solve the Differential Equation sums without any mistakes?

Practice is the best way that a student can follow to understand the sums in the Differential Equation chapter. They can use the RD Sharma Class 12th Exercise 21.2 solution material to guide them through this process.

4. What is the most common reference guide for class 12 mathematics used by the CBSE board school students?

The CBSE board school students' most preferred and used mathematics guide is that of the RD Sharma collections.

5. How many questions should a student solve for chapter 21 in class 12 mathematics?

There are 27 questions of chapter 21; mathematics asked in the textbook; along with these questions, the students must work out more additional sums to gain practice.

Sep 03, 2024

Mar 22, 2023

Get answers from students and experts

Register for Tallentex '25 - One of The Biggest Talent Encouragement Exam

As per latest 2024 syllabus. Physics formulas, equations, & laws of class 11 & 12th chapters

As per latest 2024 syllabus. Chemistry formulas, equations, & laws of class 11 & 12th chapters

Accepted by more than 11,000 universities in over 150 countries worldwide

Register now for PTE & Unlock 10% OFF : Use promo code: 'C360SPL10'. Limited Period Offer!

As per latest 2024 syllabus. Study 40% syllabus and score upto 100% marks in JEE

News and Notifications

Back to top