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In the Cartesian coordinate system, we have a family of circles with a radius 1 and these circles center at the circle

x^2+y^2=4

Mathematical, if we solve the differential equation representing the family of curves, we can find that the envelopes of the family of circles are

x^2 + y^2  = 9

and

x^2 + y^2 = 1

enter image description here

Consider the family of circles now consists of circles of the radius of 10, mathematically, we can still obtain two solutions when solving the differential equation. However, one solution should be discarded since the envelope curve represented by this solution will completely lie inside the family of circles.

How can we eliminate that unqualify solution mathematically?

Steph Fong
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  • You say (5 lines before the end line) "radius of $10$" ? I don't understand... – Jean Marie Nov 01 '20 at 23:17
  • Sorry for the confusion. So the green circles you see in the figure have a radius of 1. There are two envelopes, the two red circles. If green circles have a radius of 10, there is only one envelope geometrically. But mathematically, there are still two envelopes. How do we eliminate the unqualified one? – Steph Fong Nov 01 '20 at 23:24

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