The D.E whose solution is $\mathrm{y} = \mathrm{acos} \mathrm{x} + \mathrm{bsin} \mathrm{x} + \mathrm{xsin} \mathrm{x}$ is

$\begin{array}{l}y=a\cos x+b\sin x+x\sin x\\ y_1=-a\sin x+b\cos x+x\cos x+\sin x\\ y_2=-a\cos x-b\sin x-x\sin x+\cos x+\cos x\\ y_2=-y+2\cos x\\ \Rightarrow y_2+y=2\cos x\end{array}$