Form the differential equation of all circles which touch the x-axis at the origin.

Equation of circles with radius r which touch x axis at the origin is given by

x² + (y – r)² = r²

⇒ x² + y² + r² – 2ry = r²

⇒ x² + y² = 2ry … (1)

Differentiating both sides w.r.t. x, we get

2x + 2yy’ = 2ry’

Substituting r from (2) in (1), we get

⇒ (x² + y²)y’ = 2y(x + yy’)