包络与奇解的等价关系探究

A Discussion on the Equivalent Relationship between the Envelop and Singular Solution

摘要: 首先给出现有常微分教材中包络和奇解的几种常见定义，之后给出实例揭示二者之间存在一定关系。接下来针对包络与奇解是等价的这一结论展开研究，具体实例表明：奇解就是包络这一结论的正确性是建立在包络的宽泛定义基础之上的，在严格的包络定义下，二者并不等价。教学过程中须将二者区分对待，不能混淆。

Abstract: This paper firstly introduces several common definitions of envelop and singular solution for ordi-nary differential equations. An example reveals that there exists the relationship between the en-velop and singular solution to some extent. Then the discussion on the equivalence between envel-op and singular solution is carried out. By presenting an example we find that the validity of the conclusion that the singular solution is envelop is based on the broad definition of envelop, and un-der the strict definition of envelop the equivalence of envelop and singular solution doesn’t hold. Therefore, during the process of teaching they must be treated differently and should not be con-fused.