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It is common for class 12 students to encounter doubts while they are in the middle of doing their homework. Most of these students do not have time to attend tuition after their school hours. Hence, a proper solution guide will be of great help for them. Especially for the chapters like Differential Equation, reference material is a must. Therefore, the importance of RD Sharma Class 12th Exercise 21.9 book is inevitable.

Differential Equations Exercise 21.9 Question 1

Answer:Given: Here given that

To solve: We have to solve the given differential equation.

Hint: In homogeneous differential equation put and

Solution: We have,

Clearly since each of the functions is and homogeneous function of degree the given equation is homogeneous.

Putting and the given

Equation becomes

Integrating on both side, we get

where C=c^{2}

This is required solution.

Differential Equations Exercise 21.9 Question 2

Answer:Given:

To solve: we have to solve the given differential equation

Hint: In homogeneous differential equation put

and

Solution: we have

Cleary since each of the functions and is homogeneous function of degree the given equation is homogenous equation.

Putting and

The given equation becomes

Separating variables, we have

Integrating on both side, we get.

putting

It is required solution.

Differential Equations Exercise 21.9 Question 3

Answer:Given:

To find: we have to solve the given differential equation.

Hint: in homogeneous differential equation put and

Solution: Here,

Clearly since each of the function and is a homogeneous function of degree 2, the given equation is homogeneous

Putting and

Integrating on both side we get

This is required solution.

Differential Equations Exercise 21.9 Question 4

Answer:Given: here ,

To find: we have to find the solution of given differential equation.

Hint: in homogeneous differential equation

Put and

Solution: we have,

Clearly It is homogeneous equation

Put

So,

This is required solution.

Differential Equations Exercise 21.9 Question 5

Answer:Given: here,

To find: we have to find the solution of given differential equation.

Hint: in homogeneous differential equation

Put and

Solution: we have,

Put

This is required solution.

Differential Equations Exercise 21.9 Question 6

Answer:Given:

Hint: in homogeneous differential equation put

To solve: we have to solve the given differential Eqn.

Solution: we have

Here it is a homogeneous equation.

Put

So,

This is required solution.

Differential Equations Exercise 21.9 Question 7

Answer:Given:

To find: we have to find the solution of given differential equation.

Hint: IN homogeneous differential equation

Put and

Solution: We have

It is homogeneous equation

Put

So,

which is required solution.

Differential Equations Exercise 21.9 Question 8

Answer:Given:

To find: we have to find the solution of given differential equation

Hint: In homogeneous differential equation put and

Solution: we have,

This is a homogeneous differential equation.

Substitute and

We have,

Hence this is required solution.

Differential Equations Exercise 21.9 Question 9

Answer:Given:

To solve: we have to solve the given differential equation

Hint: In homogeneous differential equation put and

Solution: we have,

It is a homogeneous equation.

So,

This is required solution

Differential Equations Exercise 21.9 Question 10

Answer:Given:

To solve: we have to solve the given differential equation

Hint: In homogeneous differential equation put and

Solution: we have,

It is a homogeneous equation.

Put and

So,

Hence this is required solution

Differential Equations Exercise 21.9 Question 11

Answer:Given:

To find: we have to find the solution of differential equation.

Hint: In homogeneous differential equation put and

Solution: we have,

It is a homogeneous equation of degree 2.

Put and

So,

This is required solution.

Differential Equations Exercise 21.9 Question 12

Answer:Given:

To find: we have to solve the given differential equation

Hint: In homogeneous differential equation put and

Solution: we have

It is homogeneous differential equation.

Put and

So,

Differential Equations Exercise 21.9 Question 13

Answer:Given:

To solve: we have to solve given differential equation

Hint: In homogeneous differential equation put and

or put

Solution: we have,

This is homogeneous equation

Separating the variables we get

Let

This is required solution

Differential Equations Exercise 21.9 Question 14

Answer:Given:

To solve: we have to solve given differential equation

Hint: put and in homogeneous differential equation

Solution: we have,

Putand

So,

This is required solution

Differential Equations Exercise 21.9 Question 15

Answer:Given: Here,

To solve: we have to solve given differential equation

Hint: put and in homogeneous differential equation

Solution: we have,

It is homogeneous equation.

Put and

So,

Using partial fraction

Again Put

Putting the value of A and B in equation

Hence this is required solution.

Differential Equations Exercise 21.9 Question 16

Answer:Given:

To find: We have to find the solution of given differential equation.

Hint: Put and in homogeneous equation.

Solution: We have,

It is a homogeneous equation.

Put

So,

Separating the variables

This is required solution.

Differential Equations Exercise 21.9 Question 17

Answer:Given:

To solve: We have to solve the given differential equation.

Hint: Put and in homogeneous equation.

Solution:

It is a homogeneous equation

Put and

So,

Differential Equations Exercise 21.9 Question 18

Answer:Given:

Hint: Put and

Solution:

It is homogeneous equation.

Separating the variables we have,

This is required solution.

Differential Equations Exercise 21.9 Question 19

Answer:Given:

To find: We have to solve the given differential equation.

Hint: Put and

Solution: We have,

It is a homogeneous equation

Put and

So,

This is required solution.

Differential Equations Exercise 21 .9 Question 20 .

Answer:Given:

Hint:Put and

Solution: we have

It is homogeneous equation.

Putting and

So,

Separating the variables we have,

This is required solution.

Differential Equations Exercise 21.9 Question 21

Answer:Given:

To solve: we have to solve the given differential equation

Hint: Put and

Solution:

It is a homogeneous equation.

Putting and

So,

Let

This is required solution.

Differential Equations Exercise 21.9 Question 22

Answer:Given:

To Solve: We have to solve the given differential equation.

Hint: Put and

Solution: Here, given that

It is homogeneous equation.

Putting and

So, equation *becomes *

Separating the variables we get

This is required solution.

Differential Equations Exercise 21.9 Question 23

Answer:Given:

To find: We have to find the solution of given differential equation.

Hint: Put and

Solution: We have,

It is homogeneous equation.

Putting and

So,

Separating the variables, we get

This is required solution.

Differential Equations Exercise 21.9 Question 24

Answer:Given:

To solve:We have to solve the given differential equation

Hint:Put and

Put and

Solution: We have,

It is a homogeneous equation.

We put * *

* *

*So,*

* *

* *

* *

Separating the variables, we get

On integrating both sides, we get

Now, putting back the value of as , we get

Hence this is required solution.

Differential Equations Exercise 21.9 Question 25

Answer:Given:

To solve: We have to solve the given differential equation.

Hint: We have to put and

Solution: We have,

It is homogeneous equation.

Put and

So,

Integrating both sides

Put

So equation becomes

Hence this is required solution.

Differential Equations Exercise 21.9 Question 26

Answer:Given:

To solve: We have to solve the given differential equation.

Hint: In homogeneous equation we put and

Solution: Here,

It is homogeneous equation.

Put and

So,

Separating the variables and integrating we get

Comparing the coefficient of like power V.

Solving equation

Using equation

This is required solution.

Differential Equations Exercise 21.9 Question 27

Answer:Given:

To Find: We have to find the solution of the given differential equation.

Hint: Put and

Solution: We have,

It is homogeneous equation.

Put and

So,

Separating the variables and integrating both sides we get

This is required solution.

Differential Equations Exercise 21.9 Question 29

Answer:Given:

To Find: We have to find the solution of the given differential equation.

Hint: Put and

Solution: Here,

It is homogeneous equation.

Put and

So,

Separating the variables and integrating both sides we get

This is required solution.

Differential Equations Exercise 21.9 Question 30

Answer:Given:

To solve: We have to solve the given differential equation.

Hint: Put * and *

Solution: Here, we have,

It is homogeneous equation.

Put and

So,

Separating the variables and integrating both sides we get

This is required solution.

Differential Equations Exercise 21.9 Question 31

Answer:Given:

To find: we have to find the solution of given differential equation.

Hint: Put * and *

Solution: we have,

It is a homogeneous equation Put * and ** *

So,

Separating the variable and Integrating bot side we get

This is a required solution.

Differential Equations Exercise 21.9 Question 32

Answer:Given:

To solve: have to solve the given differential equation.

Hint: Put and

Solution: Here,

It is homogeneous equation

Put

So,

Separating the variable and Integrating both side we get

This is required solution.

Differential Equations Exercise 21.9 Question 33

Answer:Given:

To solve: we have to solve differential equation

Hint: Put and

Solution: we have,

Put and

So,

Separating and Integrating both side we get

Comparing the coefficient of like power of v, we get

And

So,

This is required solution.

Differential Equations Exercise 21.9 Question 34

Answer:Given:

To find: we have to find the solution of given differential equation.

Hint: we will put and

Solution: we have,

It is homogeneous equation.

Put and

So,

Separating the variables and integrating both side we get

Differential Equations Exercise 21.9 Question 35

Answer:Given:

To solve: we have to solve differential equation

Hint: Put * and *

Solution: we have,

Divided by dx both side we get

It is homogeneous equation

Put * and *

*So, *

* *

* *

* *

Separating the variables and integrating both side we get

Let

Ow equation becomes

This is required solution.

Differential Equations Exercise 21.9 Question 36 (i)

Answer:Given:

To find: we have to find the solution of given differential equation.

Hint: we will put * and *

Solution: we have,

It is homogeneous equation.

Put * and *

*So,*

* *

Integrating both side we get

Putting

.......(ii)

It is given that y(1)=0 i.e when x=1 , y=0

Putting x=1 , y=0 in (ii) we get

Putting value of c in equation (ii) we get

This is required solution.

Differential Equations Exercise 21.9 Question 36 (ii)

Answer:Given:

To find: we have to find the solution of given differential equation.

Hint: we will put and

Solution: we have,

......(i)

It is homogeneous equation.

Put and

So,

Separating the variables and integrating both side we get

Putting

....(ii)

it is given that when

Putting , in we get

Putting value of c in equation (ii) we get

This is required solution.

Differential Equations Exercise 21.9 Question 36 (iii)

Answer:Given:

To find: we have to find the solution of given differential equation.

Hint: we will put and

Solution: we have,

....(i)

It is homogeneous equation.

Put and

So,

Separating the variables and integrating both side we get

...(ii)

It is given that y=0 when x=0

Putting y=0, x=1 in equation (ii) we get

Putting value of c in equation (ii) we get

This is required solution.

Differential Equations Exercise 21.9 Question 36 (iv)

Answer:Given:

To find: we have to find the solution of given differential equation.

Hint: we will put and

Solution: we have,

....(i)

It is homogeneous equation.

Put and

So,

Separating the variables and integrating both side we get

Putting

...(ii)

It is given that y=1 when x=!

Putting y=1, x=! in equation (ii) we get

Putting value of c in equation (ii) we get

This is required solution.

Differential Equations Exercise 21.9 Question 36 (v)

Answer:Given:

To find: we have to find the solution of given differential equation.

Hint: we will put and

Solution: we have,

...(i)

It is homogeneous equation.

Put and

So,

Separating the variables and integrating both side we get

comparing the coefficient of like powers of v,

It is given that y=0 when x=0

Putting y=0, x=1 in equation (iii) we get

Putting value of c in equation (iii) we get

This is required solution.

Differential Equations Exercise 21.9 Question 36 (vi)

Answer:Given:

To find: we have to find the solution of given differential equation.

Hint: we will put * and *

Solution: we have,

.....(i)

It is homogeneous equation.

Put * and *

*So,*

* *

Separating the variables and integrating both side we get

....(ii)

Using (ii)

....(iii)

It is given that y=! when x=1

Putting y=1, x=1 in equation (iii) we get

Differential Equations Exercise 21.9 Question 36 (vii)

Answer:Given:

To find: we have to find the solution of given differential equation.

Hint: we will put and

Solution: we have,

It is homogeneous equation.

Put and

So,

Separating the variables and integrating both side we get

Putting

....(iii)

It is given that y(1)=!

Putting y=1, x=1 in equation (ii) we get

Putting value of c in equation (ii) we get

This is required solution.

Differential Equations Exercise 21.9 Question 36 (viii)

Answer:Given:

To find: we have to find the solution of given differential equation.

Hint: we will put

Solution: we have,

.....(i)

It is homogeneous equation.

put

So,

Separating the variables and integrating both side we get

....(ii)

It is given that when

Putting , in equation (ii) we get

Putting value of c in equation (ii) we get

This is required solution.

Differential Equations Exercise 21.9 Question 36 (ix)

Answer:Given:

To find: we have to find the solution of given differential equation.

Hint: we will put

Solution: we have,

...(i)

It is homogeneous equation.

put

So,

Separating the variables and integrating both side we get

...(ii)

It is given that

Putting in equation (ii) we get

Putting value of c in equation (ii) we get

This is required solution.

Differential Equations Exercise 21.9 Question 37

Answer:Given:

To find: we have to find the solution of given differential equation.

Hint: we will put

Solution: we have,

....(i)

It is homogeneous equation.

put

So,

Separating the variables and integrating both side we get

...(ii)

It is given that when

Putting , in equation (ii) we get

Putting value of c in equation (ii) we get

This is required solution.

Differential Equations Exercise 21.9 Question 38

Answer:Given:

To find: we have to find the solution of given differential equation.

Hint: we will put

Solution: we have,

..(i)

It is homogeneous equation.

put

So,

Separating the variables and integrating both side we get

It is given that y=0 when x=1

Putting y=0, x=1 in equation (ii) we get

Putting value of c in equation (ii) we get

This is required solution.

Differential Equations Exercise 21.9 Question 39

Answer:Given:

To find: we have to find the solution of given differential equation.

Hint: we will put

Solution: we have,

...(i)

It is homogeneous equation.

put

So,

Separating the variables and integrating both side we get

Putting

...(ii)

It is given that y=1 when x=0

Putting y=1, x=0 in equation (ii) we get

Putting value of c in equation (ii) we get

This is required solution.

The 21st chapter of mathematics, Differential Equations for the class 12 students, is one of the essential portions where students need to score good marks. Exercise 21.9 in his chapter has 48 questions, including its subparts. The concepts that this exercise deals with are to solve the differential equations and to find the particular solution of the differential equations. The RD Sharma Class 12 Chapter 21 Exercise 21.9 is readily available to clarify their doubts.

There are various methods by which these sums can be solved. The RD Sharma Class 12th Exercise 21.9 book consists of all these methods to adapt the ones that they feel are easy. Additionally, there are many practice questions given apart from the ones given in the textbook. With its help, the students can practice as much as they want to be clear in the concept. As this book follows the NCERT pattern, the CBSE board students have a tremendous advantage over it.

Many experts have spent scads of time providing answers for the Class 12 RD Sharma Chapter 21 Exercise 21.9 Solution book and the other RD Sharma books. However, students of every mark category like toppers, average learners, and slow learners can find their respective way of solving the problems with the help of this book. This is possible due to each sum's shortcut and elaborated methods present in RD Sharma Class 12 Solutions Differential Equation Ex 21.9 material.

The other reason why many students use RD Sharma books is free PDF copies at the Career360 website. Everyone can access the authorized RD Sharma Class 12th Exercise 21.9 for free of cost from this website. As no payment is required to own this best set of solution books, it is used by most of the students.

Another great advantage is that the staff members use the RD Sharma Solutions Chapter 21 Ex 21.9 to pick questions for the tests and exams. Therefore, when a student practices every solution given in this book, there are higher chances for the occurrence of more marks effortlessly. Therefore, visit the career 360 website to own a copy of RD Sharma solution materials.

- 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. What is the optimal solution book to learn the differential equation concept for class 12?

The RD Sharma Class 12th Exercise 21.9 is the optimal solution book for the students to clear their doubts and learn the differential integration concepts.

2. How much should I pay to download the RD Sharma book?

The career 360 website lets visitors download any RD Sharma solution books without charging even a minimal fee. Hence, it is free if you download it from this site.

3. How many questions are given in the maths textbook for exercise 21.9?

There are 48 questions, including the subparts present in the textbook; you can use the RD Sharma Class 12th Exercise 21.9 book to refer to the solutions.

4. Is the RD Sharma solution material available in PDF format?

The RD Sharma solution books are present on the Career 360 website in PDF format to download and use as a soft copy.

5. Do the RD Sharma books follow the NCERT pattern?

Yes, the RD Sharma books follow the NCERT pattern making it strongly recommendable for the CBSE board students.

Mar 22, 2023

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