The differential equation satisfied by the system of parabolas

$y^{2}=4 a x+4 a^{2}$

differentiate with respect to $x$

$\Rightarrow 2 y \frac{d y}{d x}=4 a$

$\Rightarrow a=\left(\frac{y}{2} \frac{d y}{d x}\right)$

so, required differential equation is

$y^{2}=\left(4 \times \frac{y}{2} \frac{d y}{d x}\right)^{x+4}\left(\frac{y}{2} \frac{d y}{d x}\right)^{2}$

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

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