#### Please Solve R.D.Sharma class 12 Chapter 21  Differential Equations Exercise 21.9 Question 2 Maths textbook Solution.

Answer: $log\left ( x^{2}+y^{2} \right )+2\tan ^{-1}\frac{y}{x}=k$

Given:$\frac{dy}{dx}=\frac{y-x}{y+x}$

To solve: we have to solve the given differential equation

Hint: In homogeneous differential equation put

$y=vx$ and $\frac{dy}{dx}=v+\frac{xdv}{dx}$

Solution:    we have

$\frac{dy}{dx}=\frac{y-x}{y+x}$

Cleary since each of the functions $y-x$ and$y+x$  is homogeneous function of degree $\partial$ the given equation is homogenous equation.

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

The given equation becomes

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

Separating variables, we have

$\Rightarrow \frac{v+1}{v^{2}+1}dv=\frac{dx}{x}$

Integrating on both side, we get.

$\int \frac{v+1}{v^{2}+1}=\frac{dx}{x}$

$\Rightarrow \int \frac{v}{v^{2}+1} d v+\int \frac{1}{v^{2}+1} d v=-\int \frac{d x}{x} \\$

$\Rightarrow \frac{1}{2} \log \left|v^{2}+1\right|+\tan ^{-1} v-\log x+\log c\left[\therefore \int \frac{1 d x}{1+x^{2}}\right]=\tan ^{-1} x \\$

$\Rightarrow \frac{1}{2} \log \left|v^{2}+1\right|+2 \tan ^{-1} v=2 \log \left(\frac{c}{x}\right) \\$

putting $v=y/x$

$\Rightarrow \log \left|\frac{y^{2}}{x^{2}}+1\right|+2 \tan ^{-1}\left(\frac{y}{x}\right)=2 \log \left(\frac{c}{x}\right) \\$

$\Rightarrow \log \left|\frac{y^{2}+x^{2}}{x^{2}}+1\right|+2 \tan ^{-1}\left(\frac{y}{x}\right)=2 \log \left(\frac{c}{x}\right) \\$

$\Rightarrow \log \left(y^{2}+x^{2}\right)+2\left(x^{1}\right)+\tan ^{-1}\left(\frac{y}{x}\right)=2 \log \left(\frac{c}{x}\right) \\$

$\Rightarrow \log \left(y^{2}+x^{2}\right)+2\left(\frac{y}{x}\right)=2 \log -2 \log x+2 \log x \\$

$\Rightarrow \log \left(y^{2}+x^{2}\right)+2 \tan ^{-1}\left(\frac{y}{x}\right)=2 \log c \\$

$\Rightarrow \log \left(y^{2}+x^{2}\right)+2 \tan ^{-1}\left(\frac{y}{x}\right)=K,$            $where K=2logc$

It is required solution