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$\left(x \sin ^2 \frac{y}{x}-y\right) \mathrm{dx}+\mathrm{xdy}=0, \mathrm{y}=\frac{\pi}{4}$, when $\mathrm{x}=1$
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$\left[\mathrm{x} \sin ^2 \frac{\mathrm{y}}{\mathrm{x}}-\mathrm{y}\right] \mathrm{dx}+\mathrm{xdy}=0$
$\frac{d y}{d x}=\frac{y}{x}-\sin ^2 \frac{y}{x}$ which is homogeneous _..(i)
Put $y=v x, \therefore v+x \frac{d v}{d x}=v-\sin ^2 v$ from (i)
or $\frac{d v}{\sin ^2 v}=-\frac{d x}{x}$
Integrating $\int \frac{d v}{\sin ^2 v}=-\int \frac{d x}{x} \int \operatorname{cosec}^2 v d x$
$=-\int \frac{d x}{x}-\cot v=-\log x+c$
$\log x-\cot v=c ; \log x-\cot \frac{y}{x}=c$
Putting $x=1, y=\frac{\pi}{4} ; c=-1$
Particular sol. is: $\cot \frac{\mathrm{y}}{\mathrm{x}}-\log x=1$
$\frac{d y}{d x}=\frac{y}{x}-\sin ^2 \frac{y}{x}$ which is homogeneous _..(i)
Put $y=v x, \therefore v+x \frac{d v}{d x}=v-\sin ^2 v$ from (i)
or $\frac{d v}{\sin ^2 v}=-\frac{d x}{x}$
Integrating $\int \frac{d v}{\sin ^2 v}=-\int \frac{d x}{x} \int \operatorname{cosec}^2 v d x$
$=-\int \frac{d x}{x}-\cot v=-\log x+c$
$\log x-\cot v=c ; \log x-\cot \frac{y}{x}=c$
Putting $x=1, y=\frac{\pi}{4} ; c=-1$
Particular sol. is: $\cot \frac{\mathrm{y}}{\mathrm{x}}-\log x=1$
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