Let C denote the family of circles touching x-axis at origin. Let (0, a) be the coordinates of the centre of any member of the family where ‘a’ is an arbitrary constant.
∴    the equation of family C is
x² + (y – a)² = a²
∴   x² + y² + a² – 2 a y = a²
or x² + y² – 2 a y = 0    ...(1)
a being arbitrary constant

Differentiating w.r.t. x, we get
           
                                     ...(2)
                                                                                    [Multiplying by y]
Multiplying (1) by 
                                        ...(3)
Subtracting (3) from (2), we get, 
                     
or                  or   
which is required differential equation.