$$\begin{aligned} \quad y d y & =(x d y-y d x) \sin \left(\frac{x}{y}\right) \\ \frac{d y}{y} & =\left(\frac{x d y-y d x}{y^2}\right) \sin \left(\frac{x}{y}\right) \\ \frac{d y}{y} & =\sin \left(\frac{x}{y}\right) d\left(-\frac{x}{y}\right) \\ \Rightarrow \quad \ell n y & =\cos \frac{x}{y}+C \end{aligned}$$

$$\begin{aligned} & x(1)=\frac{\pi}{2} \Rightarrow 0=\cos \frac{\pi}{2}+C \Rightarrow C=0 \\ & \text { थny }=\cos \frac{x}{y} \\ & \text { but } y=2 \Rightarrow \cos \frac{x}{2}=\ln 2 \\ & \qquad \begin{aligned} \cos x & =2 \cos ^2 \frac{x}{2}-1 \\ & =2(\ln 2)^2-1 \end{aligned} \end{aligned}$$