Mathematics - Differential Equation Question with Solution | TestHub

$\begin{array}{rl}& ydy=(xdy-ydx)\sin \left(\frac{x}{y}\right)\\ & \frac{dy}{y}=\left(\frac{xdy-ydx}{y^2}\right)\sin \left(\frac{x}{y}\right)\\ & \frac{dy}{y}=\sin \left(\frac{x}{y}\right)d\left(-\frac{x}{y}\right)\\ & \to \ell ny=\cos \frac{x}{y}+C\end{array}$
$\begin{array}{rl}& x(1)=\frac{\pi }{2}\Rightarrow 0=\cos \frac{\pi }{2}+C\Rightarrow C=0\\ & \text{ lny }=\cos \frac{x}{y}\\ & \text{ but }y=2\Rightarrow \cos \frac{x}{2}=\ell n2\\ & \cos x=2{\cos }^2\frac{x}{2}-1\\ & =2(\ln 2{)}^2-1\end{array}$