Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Please Solve R.D.Sharma class 12 Chapter 21 Differential Equations Exercise 21.9 Question 1 Maths textbook Solution.

Please Solve R.D.Sharma class 12 Chapter 21  Differential Equations Exercise 21.9 Question 1 Maths textbook Solution.

Answers (1)

Answer: x^{2}y=\left ( y+2x \right )

Given:  Here given that x^{2}dy+y\left ( x+y \right )dx=0

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

Hint: In homogeneous differential equation put y=vx and 

Solution: We have,

  x^{2}dy+y\left ( x+y \right )dx=0

 \Rightarrow \frac{dy}{dx}=\frac{y\left (x+y \right )}{x^{2}}

Clearly since each of the functions xy+y^{2}is and x^{2} homogeneous function of degree c_{i} the given equation is homogeneous.

 Putting y=vx and \frac{dy}{dx}=v+\frac{xdv}{dx}the given

Equation becomes

\begin{aligned} &v+\frac{x d v}{d x}=-\frac{v x(x+v x)}{x^{2}} \\ &\Rightarrow v+x \frac{d v}{d x}=-\frac{v x^{2}(1+v )}{x^{2}} \\ &\Rightarrow v+\frac{x d v}{d x}=-v-v^{2} \\ &\Rightarrow x \frac{d v}{d x}=-\left(2 v+v^{2}\right) \\ &\Rightarrow \frac{d v}{v^{2}+2 v} d v=-\frac{d x}{x} \end{aligned}

Integrating on both side, we get

\begin{aligned} &\int \frac{dv}{v^{2}+2 v}=-\int \frac{d x}{x} \\ &\Rightarrow \int \frac{dv}{v^{2}-2 v+1-1}=-\int \frac{d x}{x} \\ &\Rightarrow \int \frac{dv}{(v-1)^{2}-(1)^{2}}=\int \frac{d x}{x} \\ &\Rightarrow \frac{1}{2} \log \left|\frac{v+1-1}{v+1+1}\right|=-\log x+\log c \\ &\Rightarrow \log \left|\frac{v}{v+2}\right|^{1 / 2}=-\log \left|\frac{c}{x}\right|\left[\therefore \log a-\log b=\left.\log \right|^{a} / b \mid\right] \\ &\Rightarrow\left(\frac{v}{v+2}\right)^{1 / 2}=\frac{c^{2}}{x^{2}} \end{aligned}

\begin{aligned} &\therefore y=v x \Rightarrow v=\frac{y}{x} \\ &\Rightarrow \frac{y / x}{\frac{y}{x}+2}=\frac{c^{2}}{x^{2}} \\ &\Rightarrow \frac{y}{y+2 x}=\frac{c^{2}}{x^{2}} \\ &\Rightarrow x^{2} y=c^{2}(y+2 x) \\ &\Rightarrow x^{2} y=c(y+2 x) \end{aligned}                                                                                    where C=c2

This is required solution.

                

Posted by

infoexpert21

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

Board Exam Time Table 2025 Live: UP Board exams from February 24; CBSE, BSEB date sheet soon
anam-khan

CBSE Date Sheet 2025 Live: Class 10, 12 board exam dates soon; syllabus, sample papers
anam-khan

Extra time in CBSE exams for students with type 1 diabetes: Kerala SHRC seeks report on plea

Latest Question