包络排除方法及奇解排除定理

Envelope Elimination Method and Singular Solution Exclusion Theorem

讨论包络和奇解的判断问题,通过分析c-判别曲线上切线斜率的存在性,明确p-判别曲线中p的含义,得出通解曲线族中的曲线只有在c-判别曲线上每一点处存在切线斜率时,c-判别曲线才有可能是包络,否则c-判别曲线必定不是包络,另外在求奇解时,p-判别曲线中的p必须等于p-判别曲线的斜率,否则p-判别曲线必定不是奇解.

This paper discusses criteria for the envelope and the singular solution. We obtain that a necessary condition for the c-discriminant curve to be the envelope is that any solution in the general solution set has finite tangent slope at every point on the c-discriminant curve. Similarly, a necessary condition for the p-discriminant curve to be the singular solution is that the p is equal to the tangent slope of the p-discriminant curve.