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
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?