Let \(\mathrm{x}=\mathrm{x}(\mathrm{y})\) be the solution of the differential equation \( y=\left(x-y \frac{d x}{d y}\ri…

\(ydy=(xdy-ydx)\sin \left(\frac{x}{y}\right)\\ \frac{dy}{y}=\left(\frac{xdy-ydx}{{y}^{2}}\right)\sin \left(\frac{x}{y}\right)\\ \frac{dy}{y}=\sin \left(\frac{x}{y}\right)d\left(-\frac{x}{y}\right)\\ \Rightarrow \ln y=\cos \frac{x}{y}+C\\ \text{Since},x(1)=\frac{\pi }{2}\\ \Rightarrow 0=\cos \frac{\pi }{2}+C\Rightarrow C=0\\ \Rightarrow \ln y=\cos \frac{x}{y}\\ \text{ but }y=2\\ \Rightarrow \cos \frac{x}{2}=\ln 2\\ \cos x=2{\cos }^{2}\frac{x}{2}-1=2(\ln 2{)}^{2}-1\\\)