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The differential equation for $y = a \cos x + b \sin x$ is ( a,b are arbitrary constants)

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$\frac{d^{2} y}{d x^{2}} + y = 0$

$\frac{d^{2} y}{d x^{2}} - y = 0$

$\frac{d^{2} y}{d x^{2}} + (a + b) y = 0$

$\frac{d^{2} y}{d x^{2}} + (a - b) y = 0$

answer is A.

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Detailed Solution

We have,

$\begin{array}{l} y = a \cos x + b \sin x \\ \Rightarrow \frac{d y}{d x} = - a \sin x + b \cos x \\ \Rightarrow \frac{d^{2} y}{d x^{2}} = - a \cos x - b \sin x \Rightarrow \frac{d^{2} y}{d x^{2}} = - y \end{array}$

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