To find the differential equation representing a family of circles with centers on the Y-axis, let's consider the circle with center (0,K) and radius r. The general equation for such a circle is: x2+(y−K)2=r2 Differentiate equation (i) with respect to x : 2x+2(y−K)‌
dy
dx
=0 Which simplifies to: y−K=‌
−x
dy
dx
‌‌‌ or ‌‌‌K=y+‌
x
dy
dx
Let's denote ‌
dy
dx
as y1. So, from the above, we have: y−K=‌
−x
y1
‌‌⇒‌‌K=y+‌
x
y1
Now, differentiate this equation with respect to x again: 0=1+(y−K)‌