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

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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.

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