The differential equation satisfied by the system of parabolas y ^...

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

differentiate with respect to $x$

$⇒ 2y\dfrac{dy}{dx} = 4a$

$⇒ a = \left(\dfrac{y}{2}\dfrac{dy}{dx}\right)$

so, required differential equation is

${y}^{2} = \left(4 \times \dfrac{y}{2}\dfrac{dy}{dx}\right)x + 4{\left(\dfrac{y}{2}\dfrac{dy}{dx}\right)}^{2}$

$⇒ {y}^{2}{\left(\dfrac{dy}{dx}\right)}^{2} + 2xy\left(\dfrac{dy}{dx}\right) - {y}^{2} = 0$

$⇒ y{\left(\dfrac{dy}{dx}\right)}^{2} + 2x\left(\dfrac{dy}{dx}\right) - y = 0$