1. Must the singular solution (if it exists) of ODE be the envelope of the family of general solutions?

  2. If a singular solution exists, is it sure that the C-discriminant method and p-discriminant method will not miss it? If not, how can we find a singular solution in general?

I have read several books and webpages on finding the singular solutions of an ODE. Most of them say that if a singular solution exists, then it is the envelope of the family of general solutions, and therefore can be found using the C-discriminant, p-discriminant, or the simultaneous C-p method. However, as I work on some ODE problems, I find many examples where the singular solution is not an envelope of the family of general solutions. For example, $$x dy+2y dx=0$$ The general solution is $y=Cx^{-2}$. However, $x=0$ is also a solution to this ODE. Is it called a singular solution or particular solution (since it is not tangent to any integral curve)? In either case, I think I cannot obtain this solution using the C-discriminant method or p-discriminant method. Moreover, to my understanding, $x=0$ is not the envelope of the family of general solutions.

In short, I'd like to know how I can obtain the singular solutions (and particular solutions which are not contained in the general solutions) in general. Thank you very much for answering.

  • $\begingroup$ Did you find an answer , because I have same question and could not find anyone to help me! $\endgroup$ – MCS Oct 15 '17 at 17:19

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