The differential equation that represents all parabolas having their axis of symmetry coincident with the axis of x, is

$y y_2+y_1^2=0$

The equation that represents a family of parabolas having their axis of symmetry coincident with the axis of x is

$y^2=4 a(x-h)$           .....(i)

This equation contains two arbitrary constants, so we shall differentiate it twice to obtain a second order differential equation.

Differentiating (i) w.r.t. $x$, we get

$2 y \frac{d y}{d x}=4 a \Rightarrow y \frac{d y}{d x}=2 a$           ....(ii)

Differentiating (ii) w.r.t. $x$, we get

$y \frac{d^2 y}{d x^2}+\left(\frac{d y}{d x}\right)^2=0 \Rightarrow y y_2+y_1^2=0$

which is the required differential equation.