The differential equation representing the family of curves $y=A \cos (x+B)$, where $A$ and $B$ are arbitrary constant is :

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

eq of wave → $y=A \cos (x+B)$      .......(1)

A, B → artritrancy constant

finding differential equation to represent family of curves

differentiating (1) w.r.t (x)

$\frac{d y}{d x}=-A \sin (x+B)$        [as $\frac{d}{dx}(cos x) = -sin x$]

now differentiating again w.r.t (x)

$\frac{d^2 y}{d x^2}=-A \cos (x+B)$       [as $\frac{d}{dx}(sin x) = cos x$]

from (1) Substituting value of RHS (as $y=A \cos (x+B$)

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

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