\(\text { If } y=\left(x-y \frac{d x}{d y}\right) \sin \left(\frac{x}{y}\right) \text { if } x(1)=\frac{\pi}{2} \text { …

\(\begin{aligned}
& y=\left(x-y \frac{d x}{d y}\right) \sin \left(\frac{x}{y}\right) \\
& 1=\left(\frac{x}{y}-\frac{d x}{d y}\right) \sin \left(\frac{x}{y}\right) \\
& \frac{x}{y}=v \\
& \frac{d x}{d y}=v+y \frac{d v}{d y} \\
& 1=\left(v-\left(v+y \frac{d v}{d y}\right)\right) \sin v \\
& 1=v-v-y \frac{d v}{d y} \cdot \sin v \\
& 1=-y \frac{d v}{d y} \cdot \sin v \\
& \frac{d y}{y}=-\sin v d v \\
& \ln y=\cos v+c \\
& \ln y=\cos \frac{x}{y}+c \\
& 0=0+c \\
& c=0 \\
& \ln y=\cos \frac{x}{y} \\
& \ln 2=\cos \left(\frac{x}{2}\right) \\
\end{aligned}\)

\(\begin{aligned}
& \cos x=2 \cos ^2 \frac{x}{2}-1 \\
& =2 \ln ^2 2-1
\end{aligned}\)