The differential equation of all circles which pass through the origin and whose centre is on y-axis, is

a homogeneous differential equation a differential equation of order 1 and degree 2 a differential equation in variable separable form a differential equation reducible to variable separable form

a homogeneous differential equation

The differential equation representing the given family of circle is $\left(x^2-y^2\right) \frac{d y}{d x}=2 x y$ or, $\frac{d y}{d x}=\frac{2 x y}{x^2-y^2}$ Clearly, it is a homogeneous differential equation.