$\text{ Let }\mathrm{y}=\mathrm{y}\left(\mathrm{t}\right)\text{ be a solution of the differential equation }\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{\mathrm{xcosy}+\sin 2\mathrm{y}}\text{ is }\mathrm{x}={\mathrm{ce}}^{\text{siny }}-\mathrm{k}(1+\text{ siny })\text{ then the value of }\mathrm{k}\text{ is }$

$\frac{dx}{dy}=x\cos y+\sin 2y$

$\begin{array}{l}\frac{\mathrm{dx}}{\mathrm{dy}}+(-\text{ cosy })\mathrm{x}=2\text{ sinycosy }\\ \mathrm{I}.\mathrm{F}={\mathrm{e}}^{-\int \mathrm{cosydy}}={\mathrm{e}}^{-\sin y}\\ \text{ Solution of the D.E is }{\mathrm{xe}}^{\text{-siny }}=2\int {\mathrm{e}}^{\text{-siny }}\text{ .sinycosydy ..............(1)}\\ I\mathrm{ntegrate} \mathrm{by} \mathrm{parts} \mathrm{where} u=\mathrm{siny} \mathrm{and} v={\mathrm{e}}^{\text{-siny }} \mathrm{cosy}\\ \int e^{-\sin y}\cos ydy=\int e^{-t}dt=-e^{-t}+c=-e^{-\sin y}+c\\ \left(1\right) becomes\\ {\mathrm{xe}}^{\text{-siny }}=2\left[\sin y \int {\mathrm{e}}^{\text{-siny }} \mathrm{cosy}dy-\int \cos y \left(\int {\mathrm{e}}^{\text{-siny }}\mathrm{cosy}dy\right)dy\right]\\ =2\left[-\sin y {\mathrm{e}}^{\text{-siny }} +\int {\mathrm{e}}^{\text{-siny }} \mathrm{cosy}dy \right]\\ =2\left[-\sin y {\mathrm{e}}^{\text{-siny }} -{\mathrm{e}}^{\text{-siny }}\right]\\ {\mathrm{xe}}^{\text{-siny }}=-2{\text{ sinye }}^{\text{-siny }}-2{\mathrm{e}}^{\text{-siny }}+\mathrm{c}\\ \to \mathrm{x}=-2\sin \mathrm{y}-2+{\mathrm{ce}}^{\mathrm{siny}}={\mathrm{ce}}^{\mathrm{siny}}-2(1+\sin \mathrm{y})\\ \to \mathrm{k}=2\end{array}$