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If $\frac{d y}{d x}=\frac{y+x \tan \frac{y}{x}}{x}$, then $\sin \left(\frac{y}{x}\right)=$
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The correct answer is:
$c x$
We have, $\frac{d y}{d x}=\frac{y+x \tan \frac{y}{x}}{x} \quad \text{...(i)}$
Given, differential equation is in homogeneous form.
$\therefore$ put $y=v x$ in Eq. (i), we get
$v+x \frac{d v}{d x}=v+\tan v \Rightarrow \frac{1}{\tan v} d v=\frac{d x}{x}$
Taking integration on both sides, we get $\log (\sin v)=\log x+\log c$
$$
\Rightarrow \log \frac{\sin v}{x}=\log c \Rightarrow \sin \left(\frac{y}{x}\right)=x c
$$
Given, differential equation is in homogeneous form.
$\therefore$ put $y=v x$ in Eq. (i), we get
$v+x \frac{d v}{d x}=v+\tan v \Rightarrow \frac{1}{\tan v} d v=\frac{d x}{x}$
Taking integration on both sides, we get $\log (\sin v)=\log x+\log c$
$$
\Rightarrow \log \frac{\sin v}{x}=\log c \Rightarrow \sin \left(\frac{y}{x}\right)=x c
$$
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