#### Provide Solution For  R.D. Sharma Maths Class 12 Chapter 21 Differential Equations Exercise 21.9 Question 23 Maths Textbook Solution.

Answer:$|y\sin \left ( \frac{y}{x} \right )|=c$

Given:$\frac{y}{x} \cos \left(\frac{y}{x}\right) d x-\left\{\frac{x}{y} \sin \left(\frac{y}{x}\right)+\cos \left(\frac{y}{x}\right)\right\} d y=0$

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

Hint: Put $y=vx$ and $\frac{dy}{dx}=v+x\frac{dv}{dx}$

Solution: We have,

\begin{aligned} &\frac{y}{x} \cos \left(\frac{y}{x}\right) d x-\left\{\frac{x}{y} \sin \left(\frac{y}{x}\right)+\cos \left(\frac{y}{x}\right)\right\} d y=0 \\ &\Rightarrow \frac{d y}{d x}=\frac{\frac{y}{x} \cos \left(\frac{y}{x}\right)}{\frac{x}{y} \sin \left(\frac{y}{x}\right)+\cos \left(\frac{y}{x}\right)} \end{aligned}

It is homogeneous equation.

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

So,$v+x \frac{d v}{d x}=\frac{\frac{v x}{x}-\cos \left(\frac{v x}{x}\right)}{\frac{x}{v x} \sin \left(\frac{v x}{x}\right)+\cos \left(\frac{v x}{x}\right)} \\$

$\Rightarrow v+x \frac{d v}{d x}=\frac{v \cos x}{\frac{1}{v} \sin v+\cos v} \\$

$\Rightarrow v+x \frac{d v}{d x}=\frac{v^{2} \cos x}{\sin v+v \cos v} \\$

$\Rightarrow x \frac{d v}{d x}=\frac{v^{2} \cos x}{\sin v+v \cos v}-v \\$

$\Rightarrow x \frac{d v}{d x}=\frac{v^{2} \cos v-v \sin v-v^{2} \cos v}{\sin v+v \cos v}-v \\$

$\Rightarrow x \frac{d v}{d x}=\frac{-v \sin v}{\sin v+v \cos v}$

Separating the variables, we get

$\frac{\sin v+v \cos v}{v \sin v} d v=-\frac{d x}{x} \\$

$\Rightarrow \int\left(\frac{1}{v}+\cot v\right) d v=-\int \frac{d x}{x} \\$

$\Rightarrow \log v+\log |\sin v|=-\log x+\log c \\$

$\Rightarrow \log |v \sin v|=\log \frac{c}{x}\left[\therefore \log x+\log y=\log x y \text { and } \log x-\log y=\log \frac{x}{y}\right] \\$

$\Rightarrow|v \sin v|=\frac{c}{x} \\$

$\Rightarrow\left|x\left(\frac{y}{x}\right) \sin \left(\frac{y}{x}\right)\right|=c\left[\therefore v=\frac{y}{x}\right] \\$

$\Rightarrow\left|y \sin \left(\frac{y}{x}\right)\right|=c$

This is required solution.