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RD Sharma Class 12 Exercise 21.6 Differential Equation Solutions Maths - Download PDF Free Online

RD Sharma Class 12 Exercise 21.6 Differential Equation Solutions Maths - Download PDF Free Online

Edited By Lovekush kumar saini | Updated on Jan 08, 2022 11:38 AM IST

The Class 12 students need a proper guide to clarify their doubts on any concepts at their home. To make this process simpler, especially for the students who work out mathematical sums on their own for chapter 21, the RD Sharma Class 12th Exercise 21.6 solution book is handy. It is pretty standard that Differential Equation is a challenging portion. Yet with the help of authorized solution books, the students can face it easily.

Differential Equations Exercise: 21.6

Differential equations exercise 21.6 question 1

Answer: x+\frac{1}{2} \log \left|1+y^{2}\right|=C
Given: \frac{d y}{d x}+\frac{1+y^{2}}{y}=0
Hint: Separate y and x and then integrate both sides.
Solution:
\begin{aligned} &\begin{aligned} \frac{d y}{d x}+\frac{1+y^{2}}{y} &=0 \\ \frac{d y}{d x} &=-\left(\frac{1+y^{2}}{y}\right) \end{aligned}\\ &\left(\frac{y}{1+y^{2}}\right) d y=-d x\\ &\text { [integrate both side] }\\ &\int\left(\frac{y}{1+y^{2}}\right) d y=-\int d x \end{aligned}
\begin{aligned} &\text { Let } 1+y^{2}=t\\ &\text { [differentiate with respect to } y \text { ] }\\ &\begin{aligned} 2 y d y &=d t \\ & \frac{1}{2} \int \frac{1}{t} d t=-\int d x \\ & \frac{1}{2} \log |t|=-x+C \\ & \frac{1}{2} \log \left|1+y^{2}\right|+x=C \end{aligned} \end{aligned}

Differential equations exercise 21.6 question 2

Answer: x=\frac{y^{2}}{2}-\frac{1}{2} \log \left|y^{2}+1\right|+C
Given: \frac{d y}{d x}=\frac{1+y^{2}}{y^{3}}
Hint: Use integration by parts method
Solution: \frac{d y}{d x}=\frac{1+y^{2}}{y^{3}}
\frac{y^{3}}{1+y^{2}} d y=d x
On dividing \frac{y^{3}}{1+y^{2}} we will write it as y-\frac{y}{y^{2}+1} \text { i.e., } \mathrm{Q}+\frac{R}{D}

\int\left(y-\frac{y}{y^{2}+1}\right) d y=\int d x \; \; \; \; \; \; \; \; [integrate \: both\: sides]
\begin{aligned} &\int y d y-\int \frac{y}{y^{2}+1} d y=\int d x\\ &\frac{y^{2}}{2}-\frac{1}{2} \int \frac{2 y}{y^{2}+1} d y=x \end{aligned}
\text { Let } y^{2}+1=t [ Diffrentiate with reference to y]
\begin{aligned} & 2 y d y=d t \\ &\qquad \begin{array}{c} \frac{y^{2}}{2}-\frac{1}{2} \int \frac{1}{t} d t=x \\ \\\frac{y^{2}}{2}-\frac{1}{2} \log |t|=x+C \end{array} \\ \\&\qquad \begin{array}{l} \frac{y^{2}}{2}-\frac{1}{2} \log \left|y^{2}+1\right|=x+C \end{array} \end{aligned}

Differential equations exercise 21.6 question 3

Answer: x+\cot y=C
Given: \frac{d y}{d x}=\sin ^{2} y
Hint: You must know trigonometric identities
Solution: \frac{d y}{d x}=\sin ^{2} y
\begin{aligned} &\frac{1}{\sin ^{2} y} d y=d x\\ &\operatorname{cosec}^{2} y d y=d x \quad \text { [applying integration] }\\ &\int \operatorname{cosec}^{2} y d y=\int d x \quad\: \; \; \; \; \; \; \; \; \; \left[\int \operatorname{cosec}^{2} y d y=-\operatorname{coty}+c\right]\\ &-\cot y=x+C\\ &x+\cot y=C \end{aligned}

Differential equations exercise 21.6 question 4

Answer: x+\cot y+y=C
Given: \frac{d y}{d x}=\frac{1-\cos 2 y}{1+\cos 2 y}
Hint: Use trigonometric identities to simplify and integrate
Solution
\frac{d y}{d x}=\frac{1-\cos 2 y}{1+\cos 2 y} \left[\cos 2 x=1-\sin ^{2} x=2 \cos ^{2} x-1\right]
\frac{d y}{d x}=\frac{2 \sin ^{2} y}{2 \cos ^{2} y} \left[\frac{\sin x}{\cos x}=\tan x\right]
\frac{d y}{d x}=\tan ^{2} y
\frac{1}{\tan ^{2} y} d y=d x \left[\frac{1}{\tan x}=\cot x\right]
\int \cot ^{2} y d y=\int d x \left[\cot ^{2} y=\operatorname{cosec}^{2} y-1\right]
\int\left(\operatorname{cosec}^{2} y-1\right) d y=\int d x \left[\int \operatorname{cosec}^{2} y d y=-\cot y+c\right]
\begin{aligned} &-\cot y-y+C=x \\ &-\cot y-y-x=-C \\ &C=\cot y+y+x \end{aligned}

Also see,

  • RD Sharma Solutions Class 12 Mathematics Chapter 21 Exercise 21.1

  • RD Sharma Solutions Class 12 Mathematics Chapter 21 Exercise 21.2

  • RD Sharma Solutions Class 12 Mathematics Chapter 21 Exercise 21.3

  • RD Sharma Solutions Class 12 Mathematics Chapter 21 Exercise 21.4

  • RD Sharma Solutions Class 12 Mathematics Chapter 21 Exercise 21.6

  • RD Sharma Solutions Class 12 Mathematics Chapter 21 Exercise 21.7

  • RD Sharma Solutions Class 12 Mathematics Chapter 21 Exercise 21.8

  • RD Sharma Solutions Class 12 Mathematics Chapter 21 Exercise 21.9

  • RD Sharma Solutions Class 12 Mathematics Chapter 21 Exercise 21.10

  • RD Sharma Solutions Class 12 Mathematics Chapter 21 Exercise 21.11

  • RD Sharma Solutions Class 12 Mathematics Chapter 21 Exercise FBQ

  • RD Sharma Solutions Class 12 Mathematics Chapter 21 Exercise MCQ

  • RD Sharma Solutions Class 12 Mathematics Chapter 21 Exercise CSBQ

  • RD Sharma Solutions Class 12 Mathematics Chapter 21 Exercise RE

The Class 12 syllabus consists of the Differential Equation concept as its 21st chapter. Among the eleven exercises given in this chapter, the sixth exercise, ex 21.6, has fewer sums given in the textbook. This is because there are only four sums to be solved under the concept of the differential equation. Even though the count of the sums is less, students get doubts in the middle of these sums. The RD Sharma Class 12 Chapter 21 Exercise 21.6 reference material will lend a helping hand in such circumstances.

Exercise 21.6 has only Level 1 sums; therefore, it would not be tough to solve. Yet, these questions are not enough for a student to understand the concept deeply. Consequently, you can find various other practice sums to work out on the same concept given in the RD Sharma Class 12th Exercise 21.6 book. This is because Differential Equation is a concept that will only be easier to solve with a lot of practice. Therefore, the NCERT pattern solutions in this book give another valid reason for the CBSE students to adapt it.

The students can trust the answers given in the Class 12 RD Sharma Chapter 21 Exercise 21.6 Solution as much as they trust their mathematics teacher. A committee of staff members provides all the solutions presented here with very high expertise in this domain. Therefore, not only does this book help the student in preparing for their exams, but it also lends a helping hand in solving their homework and assignments.

And the other benefit that excites most of the students is that the RD Sharma Class 12 Solutions Differential Equation Ex 21.6 book can be downloaded for free from the Career 360 website. Thus, every student can access the RD Sharma Class 12th Exercise 21.6 book without spending money as they usually do to buy other books.

The usage of RD Sharma Class 12 Solutions Chapter 21 Ex 21.6 reference material to prepare questionnaires by teachers has increased its value. In the presence of this book, you need not depend on a teacher or a tutor each time you get doubts in solving differentiation.

RD Sharma Chapter wise Solutions

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

Frequently Asked Questions (FAQs)

1. How many exercises are present in Class 12, mathematics, Chapter 21?

There are eleven exercises ex 21.1 to ex 21.11 in chapter 21, Differential Function chapter.

2. Is there a solution book for Differential Equation in the set of RD Sharma books?

The RD Sharma Class 12th Exercise 21.6 is the best solution book that students can easily understand the Differential Equations concepts. 

3. Are the RD Sharma book PDFs available for free?

Yes, you can find the RD Sharma solution book PDFs on the Career 360 website.

4. How many questions are present are in the RD Sharma solution book for exercise 21.6?

There are only four sums in the textbook, and the solutions are provided in the RD Sharma Class 12th Exercise 21.6 reference material.

5. Can I download the RD Sharma solution book to my device?

Yes, all the RD Sharma reference materials have the option that makes the students download it into their device for further use.

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