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

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

$(x-a{)}^2+(y-2{)}^2=5^2$

Differentiating w.r. to x, we get

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

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