The equation of family of circles touching the y-axis at the origin is, ‌(x±h)2+y2=h2 ⇒‌‌x2+y2+2xh=0 . . . (i) On differentiating w.r.t. x, we get 2x+2y‌
dy
dx
+2h=0 or 2x+2y⋅y′+2h=0(∵−1‌
dy
dx
=y′) ⇒‌‌2h=−(2x+2yy′) On putting the value of 2h in Eq. (i), we get ‌x2+y2−x(2x+2yy′)‌=0 ⇒‌‌x2+y2−2x2−2xyy′‌=0 ⇒‌‌2xyy′+x2‌=y2 which is the required differential equation.