If y = a sin x + ( 5 + 2 x ) cos x , then y ' ' + y =

Given:

$y=a\sin x+(5+2x)\cos x ..\left(i\right)$

Differentiating w.r.t. $x$, we get

$y^{\text{'}}=a\cos x+2\cos x-\left(5+2x\right)\sin x$

$y^{\text{'}}=\left(a+2\right)\cos x-\left(5+2x\right)\sin x$

Again differentiating w.r.t. $x$, we get

$y^{\text{'}\text{'}}=-\left(a+2\right)\sin x-\left(5+2x\right)\cos x-2\sin x ...\left(ii\right)$

Adding $\left(i\right) \& \left(ii\right)$, we get

$y^{\text{'}\text{'}}+y=a\sin x+(5+2x)\cos x -\left(a+2\right)\sin x-\left(5+2x\right)\cos x-2\sin x$

            $=-4\sin x$.