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=r2Differentiate equation (i) with respect to
x :
2x+2(y−K)‌=0Which simplifies to:
y−K=‌‌‌‌ or ‌‌‌K=y+‌Let's denote
‌ as
y1. So, from the above, we have:
y−K=‌‌‌⇒‌‌K=y+‌Now, differentiate this equation with respect to
x again:
0=1+(y−K)‌+(‌)2Substituting
y−K=‌ into this, we get:
1+(−‌)‌+y12=0This can be rearranged to:
‌1−‌+y12=0‌x‌=y1(y12+1)So, the differential equation is:
x‌=y1(y12+1)