Family y = Ax + A³ of curves will correspond to a differential equation of order

The correct option is 2.

We are given the equation for the family of curves:
$y=Ax+A^3$
where $A$ is an arbitrary constant parameter.

To find the corresponding differential equation, we first differentiate the equation with respect to $x$:
$\frac{dy}{dx}=A$

To eliminate the parameter $A$, we differentiate a second time with respect to $x$:
$\frac{d^{2}y}{d{x}^2}=0$

Since the resulting differential equation involves the second derivative, it represents a differential equation of order 2.