Let y = y(x) be the solution of the differential equation ,$\cos \mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}+2\mathrm{ysin}\mathrm{x}=\sin 2\mathrm{x}$ $\mathrm{x}\in (0,\mathrm{\pi }/2)\text{. If }\mathrm{y}(\mathrm{\pi }/3)=0\text{, then }\mathrm{y}(\mathrm{\pi }/4)=$

Given differential equation is $\cos \mathrm{x}\frac{\mathrm{dy}}{\mathrm{dx}}+2\mathrm{ysin}\mathrm{x}=\sin 2\mathrm{x}\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}}+2\tan \mathrm{xy}=2\sin \mathrm{x}$

It is linear in x. $\text{ I.F }={\mathrm{e}}^{\int 2\tan \mathrm{xdx}}={\mathrm{e}}^{2\log \sec \mathrm{x}}={\sec }^2\mathrm{x}$

$\therefore$The solution is${\mathrm{ysec}}^2\mathrm{x}=\int 2\sin {\mathrm{xsec}}^2\mathrm{xdx}=2\int \tan \mathrm{xsec}\mathrm{xdx}=2\sec \mathrm{x}+\mathrm{c}$+

$\begin{array}{l}y(\pi /3)=0\Rightarrow 0=4+c\Rightarrow c=-4.\\ \therefore y{\sec }^{2}x=2\sec x-4\Rightarrow y=2\cos x-4{\cos }^{2}x.\text{ Now }y\left(\frac{\pi }{4}\right)=2\left(\frac{1}{\sqrt{2}}\right)-4{\left(\frac{1}{\sqrt{2}}\right)}^2=\sqrt{2}-2\end{array}$