Must the singular solution of ODE be the envelope of the family of general solutions?

Question

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.

Answer 1

Though I don't know the answer to your questions 1 & 2, your example requires some reconsideration:

No, actually $x=0$ and $y=0$ are particular solutions, as if we choose arbitrary constant $c=0$ then we get $y=0$ as a solution. Also $x^2 = A y$ is a general solution, so from here the choice of arbitrary constant $A=0$ yields $x=0$ as a particular solution.

Also the given differential equation does not have any singular solution.