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The D.E whose solution is $y = acos x + bsin x + xsin x$ is

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$y_{2} + y = 2 \sin x$

$y_{2} + y = 2 \cos x$

$y_{2} + y = \sin x$

$y_{2} + y = \cos x$

answer is D.

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Detailed Solution

$\begin{array}{l} y = a \cos x + b \sin x + x \sin x \\ y_{1} = - a \sin x + b \cos x + x \cos x + \sin x \\ y_{2} = - a \cos x - b \sin x - x \sin x + \cos x + \cos x \\ y_{2} = - y + 2 \cos x \\ \Rightarrow y_{2} + y = 2 \cos x \end{array}$

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