Degenerate ordinary differential equation

A degenerate ordinary differential equation has a highest-derivative coefficient that vanishes somewhere in its domain. Its solutions can have lower regularity at that point than in the interior. A boundary value problem normally imposes endpoint values by continuous extension while requiring the differential equation in the interior. Requiring it pointwise at a degenerate endpoint with all derivatives finite can impose an extra condition and destroy existence. In a square-root-degenerate endpoint layer, the product of a vanishing square-root coefficient with a diverging second derivative has a finite nonzero interior limit.