【摘要】 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。更多还原

【Abstract】 According to the fundamental theory of differential geometry,an envelope of a family of curves with one parameter satisfies its C-discriminant. This result is often applied to study singular solutions of ordinary differential equations. However,there are counterexamples for this result. 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 satisfied C-discriminant are given so that its envelope,and an applicable method of getting singular solutions of ordinary differential equations is proposed.更多还原