Form the differential equation representing the family of curves $y=a \sin (x+b),$ where $a$ and $b$ are arbitrary constants.

(A) Given the equation of the family of curves:
$y = a \sin(x + b)$ --- $(1)$
Differentiating equation $(1)$ with respect to $x$:
$\frac{dy}{dx} = a \cos(x + b)$ --- $(2)$
Differentiating equation $(2)$ with respect to $x$ again:
$\frac{d^2y}{dx^2} = -a \sin(x + b)$ --- $(3)$
From equation $(1)$,we know that $a \sin(x + b) = y$. Substituting this into equation $(3)$:
$\frac{d^2y}{dx^2} = -y$
Rearranging the terms,we get the required differential equation:
$\frac{d^2y}{dx^2} + y = 0$