The equation of family of circles touching y-axis at origin is
(x – h)² + y² = h² where h is arbitrary constant    ...(1)
[∵ if (h, 0) is centre of any member of family, then its radius = h]
Differentiating (1) w.r.t.x,  
⇒ x – h + y y₁= 0 ⇒ h = x + y y₁
Putting value of h in (1), we get,
(x – x – y y₁)² + y² = (x + y y₁)² ⇒ y² y₁² + y² = y² + 2 x y y₁ + y² y₁²
⇒ x² – y² + 2 x y y₁ = 0 , which is the required differential equation.