If \(y = x\sin x\), then \(\displaystyle\frac{d^{2}y}{dx^{2}} + y\) is equal to
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(2\cos x\)
By the product rule:
\( \displaystyle \frac{dy}{dx} = \sin x + x\cos x \)
\( \displaystyle \frac{d^{2}y}{dx^{2}} = \cos x + \cos x - x\sin x = 2\cos x - x\sin x \)
Adding \(y = x\sin x\):
\( \displaystyle \frac{d^{2}y}{dx^{2}} + y = 2\cos x - x\sin x + x\sin x = 2\cos x \)
The value 0 would hold for \(y = \sin x\) alone; the factor x adds the extra term \(2\cos x\) through the product rule.