RD Sharma Class 12 Exercise 21.4 Differential Equation Solutions Maths - Download PDF Free Online

# RD Sharma Class 12 Exercise 21.4 Differential Equation Solutions Maths - Download PDF Free Online

Edited By Lovekush kumar saini | Updated on Jan 24, 2022 04:20 PM IST

Class 12 students spend most of their time preparing for the public exams. The right solution book is required to clarify the doubts of the students in every subject, especially mathematics. Chapters like the Differential Equations are very challenging for most of the students. The RD Sharma Class 12th Exercise 21.4 reference book is very helpful for the students to complete their homework and assignments. It also lends them a helping hand in preparing for their daily tests and public exams.

## Differential Equations Excercise: 1.3

Differential equations excercise 21.4 question 1

Answer: $y=\log x$ is the solution of given function.
Hint:
Just differentiate the function & then put value
Given:
$y=\log x$ is the function.
Solution:
Differentiate with respect to x
\begin{aligned} &\Rightarrow \frac{d y}{d x}=\frac{d}{d x}(\log x) \\ &\Rightarrow \frac{d y}{d x}=\frac{1}{x} \\ &\Rightarrow x \frac{d y}{d x}=1 \end{aligned}
Hence the function satisfies the equation.
Also, when
\begin{aligned} &x=1 \\ &y=\log 1=0 \end{aligned}
Thus $y(1)=0$ satisfies the initial value problem.

Differential equation exercise 21.4 question 2

Answer: $y=e^{x}$ is the solution of given function
Hint: Differentiate the function with respect to x
Given:
$y=e^{x}$ is the function.
Solution:
Differentiate with respect to x
\begin{aligned} &\Rightarrow \frac{d y}{d x}=\frac{d e^{x}}{d x} \\ &\Rightarrow \frac{d y}{d x}=e^{x} \\ &\Rightarrow \frac{d y}{d x}=y\left[\because y=e^{x}\right] \end{aligned}
Thus, $y\left(e^{x}\right)$ satisfies the equation.
Now , When
\begin{aligned} &x=0 \\ &y=e^{0}=1 \end{aligned}
Thus, $\lambda(0)=1$ also satisfies the given equation.

Differential equation excercise 21.4 question 3

Answer: $y=\sin x$ is the solution of given function.
Hint:
Take derivative of the function and check that the solution is satisfying or not.
Given:
$y=\sin x$ is the function.
Solution:
Differentiate with respect to x
\begin{aligned} &\Rightarrow \frac{d y}{d x}=\frac{d}{d x}(\sin x)\\ &\Rightarrow \frac{d y}{d x}=\cos x \cdots(i)\\ &\text { Differentiating eq(i) }\\ &\Rightarrow \frac{d^{2} y}{d x^{2}}=-\sin x\\ &\left.\Rightarrow \frac{d^{2} y}{d x^{2}}=-y \quad \because \because y=\sin x \text { is given }\right]\\ &\Rightarrow \frac{d^{2} y}{d x^{2}}+y=0 \end{aligned}
Thus , $y=\sin x$ satisfies the initial value problem
Now,
\begin{aligned} y(0) &=0 \\ x &=0 \\ \text { then } y &=\sin 0=0 \end{aligned}
Thus $y(0)=0$ also satisfies initial value problem
\begin{aligned} &\text { Now, }\\ &\begin{aligned} y^{\prime}(0) &=1 \\ i e, y^{\prime} &=\cos x \\ & \therefore x=0, y^{\prime}=\cos 0=1 \end{aligned} \end{aligned}

Thus $y^{\prime}(0)=1$also satisfies initial value problem.

Differential equations excercise 21.4 question 4

Answer: $y=e^{x}+1$ is the solution of given function
Hint:
Take the differentiation of the function $y=e^{x}+1$
Given:
$y=e^{x}+1$ is the function.
Solution:
Differentiating with respect to x
$\Rightarrow \frac{d y}{d x}=e^{x} \cdots(i)$
Again differentiating eq(i)
\begin{aligned} &\Rightarrow \frac{d^{2} y}{d x^{2}}=e^{x} \\ &\Rightarrow \frac{d^{2} y}{d x^{2}}=\frac{d y}{d x} \quad\left[\because \frac{d y}{d x}=e^{x}\right] \\ &\Rightarrow \frac{d^{2} y}{d x^{2}}-\frac{d y}{d x}=0 \end{aligned}
Thus, $y=e^{x}+1$ satisfies the function.
Now,
When $x=0$
\begin{aligned} y &=e^{0}+1 \\ &=1+1 \\ &=2 \end{aligned}
Now,
When $x=0$
$y^{\prime}=e^{x}=e^{0}=1$
Thus both $y(0)=2 \text { and } y^{\prime}(0)=1$ satisfies the initial value problem.

Differential equations exercise 21.4 question 5

Answer: $y=e^{-x}+2$ is the solution of given function
Hint:
Differentiate the function and then obtain the value and satisfy initial value of problem.
Given:
$y=e^{-x}+2$ is the function.
Solution:
Differentiating on both sides with respect to x
$\Rightarrow \frac{d y}{d x}=-e^{-x}$
\begin{aligned} &\text { For } e^{-x}=y-2 \quad\left[\therefore y=e^{-x}+2\right] \\ &\Rightarrow \frac{d y}{d x}=-(y-2) \\ &\Rightarrow \frac{d y}{d x}=-y+2 \\ &\Rightarrow \frac{d y}{d x}+y=2 \end{aligned}
Thus,
$y=e^{-x}+2$ satisfies the equation
\begin{aligned} &\text { Now, When }\\ &x=0\\ &y=e^{-(0)}+2\\ &y=1+2\\ &y=3 \end{aligned}
Thus, $y(0)=3$ solves initial value problem.

Differential equations excercise 21.4 question 6

Answer: $y=\sin x+\cos x$ is the solution of given function
Hint:
Differentiate the function and then place value of x as per given.
Given:
$y=\sin x+\cos x$ is the function.
Solution:
Differentiating on both sides with respect to x
$\Rightarrow \frac{d y}{d x}=\cos x-\sin x \cdots(i)$
\begin{aligned} &\text { Differentiating eq(i) w.r.t } x\\ &\Rightarrow \frac{d^{2} y}{d x^{2}}=-\sin x-\cos x\\ &\Rightarrow \frac{d^{2} y}{d x^{2}}=-(\sin x+\cos x)\\ &\Rightarrow \frac{d^{2} y}{d x^{2}}=-y\\ &\Rightarrow \frac{d^{2} y}{d x^{2}}+y=0 \end{aligned}
Thus, $y=\sin x+\cos x$ satistfies the equation
\begin{aligned} &\text { When, } x=0\\ &\begin{aligned} y &=\sin 0+\cos 0 \\ &=0+1 \\ &=1 \end{aligned} \end{aligned}
$when, x=0.$
\begin{aligned} y^{\prime} &=\sin 0-\cos 0 \\ &=-1 \end{aligned}
$\therefore \text { For both } y(0)=1 \text { and } y^{\prime}(0)=-1 \text { the functions satisfies the initial value problem }$.

Differential equations exercise 21.4 question 7

Answer: $y=e^{x}+e^{-x}$ is the solution of given function
Hint:
Differentiate the function and then substitute
Given:
$y=e^{x}+e^{-x}$ is the function.
Solution:
Differentiating $y=e^{x}+e^{-x}$ with respect to x
\begin{aligned} &\Rightarrow \frac{d y}{d x}=e^{x}-e^{-x} \cdots(i)\\ &\text { Again differentiating eq(i) }\\ &\Rightarrow \frac{d^{2} y}{d x^{2}}=e^{x}+e^{-x}\\ &\Rightarrow \frac{d^{2} y}{d x^{2}}=y\\ &\Rightarrow \frac{d^{2} y}{d x^{2}}-y=0 \end{aligned}
Thus $y=e^{x}+e^{-x}$ satisfies the equation
Now, when $x=0$
\begin{aligned} \mathrm{y} &=\boldsymbol{e}^{0}+e^{-(0)} \\ &=1+1=2 \end{aligned}
Now, when $x=0$
\begin{aligned} \mathrm{y}^{\prime} &=e^{0}-e^{-(0)} \\ &=1-1=0 \end{aligned}
Thus both $y(0)=2 \text { and } y^{\prime}(0)=0$ satisfies the equation.

Differential equations excercise 21.4 question 8

$y=e^{x}+e^{2 x}$ is the solution of given function
Hint:
Differentiate the function
Given:
$y=e^{x}+e^{2 x}$ is the given function
Solution:
Differentiating with respect to x
$\Rightarrow \frac{d y}{d x}=e^{x}+e^{2 x} \cdots(i)$
Differentiating with respect to x
\begin{aligned} &\Rightarrow \frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left(e^{x}+2 e^{2 x}\right) \\ &\Rightarrow \frac{d^{2} y}{d x^{2}}=e^{x}+2\left(2 e^{2 x}\right) \\ &\Rightarrow \frac{d^{2} y}{d x^{2}}=e^{x}+4 e^{2 x} \cdots(i i) \end{aligned}
Put value (i) and (ii) in different equations
\begin{aligned} \frac{d^{2} y}{d x^{2}} &-3 \frac{d y}{d x}+2 y=0 \\ L H S &=\frac{d^{2} y}{d x^{2}}-3 \frac{d y}{d x}+2 y \\ &=\left(e^{x}+4 e^{2 x}\right)-3\left(e^{x}+2 e^{2 x}\right)+2\left(e^{x}+e^{2 x}\right) \\ &=e^{x}+4 e^{2 x}-3 e^{x}-6 e^{2 x}+2 e^{x}+2 e^{2 x} \\ &=0 \\ &=R H S \end{aligned}

Thus,
$y=e^{x}+e^{2 x}$ satisfies the diffrential equation
Now,when $x=0$
\begin{aligned} \mathrm{y} &=e^{0}+e^{2(0)} \\ &=1+e^{0} \\ &=1+1 \\ &=2 \end{aligned}
Now,when $x=0$
\begin{aligned} y^{\prime} &=e^{0}+2 e^{2(0)} \\ &=1+2(1) \\ y^{\prime} &=3 \end{aligned}
Thus, $y(0)=2 \text { and } y^{\prime}(0)=3$ satisfies the initial value problem.

Differential equation exercise 21.4 question 9

Answer: $y=x e^{x}+e^{x}$ is the solution of given function
Hint:
Differentiate the function
Given:
$y=x e^{x}+e^{x}$ is the function.
Solution:
Differentiate with respect to x
\begin{aligned} &\Rightarrow \frac{d y}{d x}=\frac{d}{d x}\left(x e^{x}+e^{x}\right) \\ &\Rightarrow \frac{d y}{d x}=\frac{d}{d x}\left(x e^{x}\right)+\frac{d}{d x}\left(e^{x}\right) \\ &\Rightarrow \frac{d y}{d x}=x e^{x}+e^{x}+e^{x} \\ &\Rightarrow \frac{d y}{d x}=x e^{x}+2 e^{x} \cdots(i) \end{aligned}
Differentiate (i) with respect to x
\begin{aligned} &\Rightarrow \frac{d^{2} y}{d x^{2}}=\frac{d}{d x}\left(x e^{x}+2 e^{x}\right) \\ &\Rightarrow \frac{d^{2} y}{d x^{2}}=x e^{x}+e^{x}+2 e^{2 x} \\ &\Rightarrow \frac{d^{2} y}{d x^{2}}=x e^{x}+3 e^{2 x} \cdots(i i) \end{aligned}
\begin{aligned} &\text { Put value (i) and (ii) in given equation }\\ &\begin{aligned} \frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}+y &=0 \\ L H S &=\frac{d^{2} y}{d x^{2}}-2 \frac{d y}{d x}+y \\ &=\left(x e^{x}+3 e^{2 x}\right)-2\left(x e^{x}+2 e^{x}\right)+\left(x e^{x}+e^{x}\right) \\ &=x e^{x}+3 e^{x}-2 x e^{x}-4 e^{x}+x e^{x}+e^{x} \\ &=0 \\ &=R H S \end{aligned} \end{aligned}
Thus,$y=x e^{x}+e^{x}$ satisfies the equation
Now, when $x = 0$
\begin{aligned} \mathrm{y} &=0 e^{0}+e^{0} \\ &=0+1 \\ &=1 \end{aligned}
Now, when $x = 0$
\begin{aligned} \mathrm{y}^{\prime} &=0 e^{0}+2 e^{0} \\ &=0+2 \\ \mathrm{y}^{\prime} &=2 \end{aligned}
Thus $y(0)=1 \text { and } y^{\prime}(0)=2$ satisfies the equation and initial problem

The topics in class 12 mathematics, chapter 21, Differential Equations, are complex when compared to the other concepts in the book. There are eleven exercises in this chapter, ex 21.1 to ex 21.11. The fourth exercise, ex 21.4, consists of concepts with various methods in which the Differential Equations can be solved. The RD Sharma Class 12 Chapter 21 Exercise 21.4 book contains solved sums for the nine questions given in this exercise.

Most of the teachers pick the questions from the RD Sharma Class 12th Exercise 21.4 while conducting tests and exams. Therefore, the students who practice with the RD Sharma Solutions tend to score more marks when compared to their classmates. Moreover, various experts in the respective fields provide the answers in these solution books and are verified to check their correctness. Therefore, the students need not worry about the accuracy of the given answers.

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Make the best choice of using the RD Sharma reference materials from Day 1 of your preparation for the exams. Utilize the chance to attain this set of solution books for free and own a copy of the RD Sharma Class 12 Solutions Chapter 21 Ex 21.4 from the Career 360 website.

RD Sharma Chapter wise Solutions

1. What is so special about the RD Sharma solutions books?

The RD Sharma books are the most recommended set of solution materials among the students and the teachers.

2. The questions for the public exams are also picked from the practice questions section given in this book.

How many sums are solved in the RD Sharma solution guide for Exercise 21.4 given in the textbook?

There are around nine questions present in Exercise 21.4 given in the class 12 mathematics textbook. You can find the solved sums for all these questions in the RD Sharma Class 12th Exercise 21.4 book.

3. Can the students use the RD Sharma books to prepare for their public exams?

The students can refer to this best set of solution books for their public exam preparations, JEE mains, and other entrance exam preparations.

4. What is the best source to download the RD Sharma Solutions book?

5. Which reference guide would help the students to solve the sums in the Differential Equations chapter with ease?

The RD Sharma Class 12th Exercise 21.4 reference book consists of many possible methods to solve the sums. Therefore, this is the best option for the students to clarify their doubts in Differential equation sums.

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