The differential equation of the family of parabolas having vertex at the origin and the y-axis as the axis of symmetry, is

$x y_1-2 y=0$

The equation of the family of parabolas having vertex at the origin and the y-axis as the axis of symmetry is

$y=a x^2$

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

$\Rightarrow \frac{d y}{d x}=\frac{2 y}{x}$       [∵ $y=a x^2 \Rightarrow a=y / x^2$]

$\Rightarrow x y_1-2 y=0$

This is the required differential equation.