The differential equation of family of circles with fixed radius 5 units and centre on the line $y=2$, is

$(y-2)^2 y^{\prime 2}=25-(y-2)^2$

Let $(a, 2)$ be the centre of the circle, where 'a' is a variable. Then, the equation of the family of circles is

$(x-a)^2+(y-2)^2=5^2$             ......(i)

Differentiating w.r. to $x$, we get

$2(x-a)+2(y-2) \frac{d y}{d x} \Rightarrow x-a=-(y-2) y_1$

Substituting this value of $(x-a)$ in (i), we get

$(y-2)^2 y_1^2=25-(y-2)^2$

as the required differential equation.