The differential equation for $y=A \cos (\alpha x)+B \sin (\alpha x)$, where A and B are arbitrary constants is:

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

The correct answer is Option (1) → $\frac{d^2 y}{d x^2}+\alpha^2 y=0$

No. of arbitrary constants = 2

So order → 2

So $\frac{dy}{dx}=α(-A\sin(αx)+\cos(αx))$

or $\frac{d^2y}{dx^2}=α^2(-A\sin(αx)-\cos(αx))$

$\frac{d^2y}{dx^2}=-α^2y$ or $\frac{d^2y}{dx^2}+α^2y=0$