Differential Equations

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)‌

d2y

dx2

+(‌

dy

dx

)2
Substituting y−K=‌

−x

y1

into this, we get:
1+(−‌

x

y1

)‌

d2y

dx2

+y12=0
This can be rearranged to:
‌1−‌

x d2y dx2

y1

+y12=0
‌x‌

d2y

dx2

=y1(y12+1)
So, the differential equation is:
x‌

d2y

dx2

=y1(y12+1)