The differential equation representing the family of curves $y=asin (x+b)$, where a and b are arbitrary constants is :

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

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

The correct answer is Option (1) → $\frac{d^2y}{dx^2}+y = 0 $ $y=a\sin (x+b)$ No. of arbitrary constants = 2 degree of equation will be 2 so $\frac{dy}{dx}=a\cos(x+b)$ $\frac{d^2y}{dx^2}=-a\sin(x+b)=-y$ so $\frac{d^2y}{dx^2}+y=0$