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Explain solution for  RD Sharma Class 12 Chapter 21 Differential Equation Exercise Multiple Choice Question Question 21 maths textbook solution.

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Answer : \text { (b) } \sin \frac{y}{x}=C x

Hint : Put \frac{y}{x}=v

Given : x \frac{d y}{d x}=y+x \tan \frac{y}{x}

Explanation : x \frac{d y}{d x}=y+x \tan \frac{y}{x}                          ....(i)

\begin{aligned} &\text { Let } y=v x \\ &\Rightarrow \frac{d y}{d x}=v+x \frac{d v}{d x} \end{aligned}

Put in (i) we have

\begin{aligned} &\Rightarrow v+x \frac{d v}{d x}=v+\tan v \\ &\Rightarrow x \frac{d v}{d x}=\tan v \\ &\Rightarrow \frac{d v}{\tan v}=\frac{d x}{x} \end{aligned}

\Rightarrow \cot v d v=\frac{d x}{x}

Integrate both sides

\begin{aligned} &\Rightarrow \int \cot v d v=\int \frac{d x}{x} \\ &\Rightarrow \log \sin v=\log x+\log C \\ &\Rightarrow \log \sin v=\log C x \end{aligned}

\begin{aligned} &\Rightarrow \sin v=C x \\ &\Rightarrow \sin \frac{y}{x}=C x \end{aligned}

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