关于单参数曲线族的包络与C-判别式的探讨

Study on envelope of a family of curves with one parameter and its C-discriminant

根据微分几何基本理论,单参数曲线族k(C)的包络必满足C-判别式。常微分方程中也利用此结论研究方程的奇解,然而,上述结论存在反例。本文先在一个较一般情况下得到包络满足一个较弱条件的结论。然后,利用此结论,得到在一些较易验证且不失一般性的条件之下包络满足的条件。最后,给出包络满足C-判别式的条件,并给出了求常微分方程奇解的一个实用方法。

A weak condition satisfied by the envelope is given under certain general assumptions. Based on it, some conditions on the family of curves, which can be easily tested, are given. Finally, the conditions that satisfied C-discfiminant are given so that its envelope, and an applicable method of obtaining singular solutions of ordinary differential equations is proposed.