The solution of $\frac{\mathrm{dy}}{\mathrm{dx}}+\frac{y \mathrm{cosx}+\mathrm{siny}+\mathrm{y}}{\mathrm{sinx}+\mathrm{x} \mathrm{cosy}+\mathrm{x}}=0$ is

$\begin{array}{l}\frac{dy}{dx}=\frac{-y\cos x-\sin y-y}{\sin x+x\cos y+x}\\ \Rightarrow \sin xdy+x\cos ydy+xdy=-y\cos xdx-\sin ydx-ydx\\ \Rightarrow \sin xdy+\sin ydx+x\mathrm{cosy}dy+ycosxdx+xdy+ydx=0\\ \Rightarrow (\sin xdy+y\cos xdx)+(\sin ydx+x\mathrm{cosy}dy)+(xdy+ydx)=0\\ \Rightarrow \int D\left(y\sin x\right)+\int D\left(x\sin y\right)+\int D\left(xy\right)=0\\ \Rightarrow y\sin x+x\sin y+xy=k\end{array}$