Must the singular solution (if it exists) of ODE be the envelope of the family of general solutions?
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.
Though I don't know answer of your questions 1 &2, you're example require 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 a solution. Also x^2 = A y ,is general solution so from here particular choice of arbitrary constant A ,we get x=0 a particular solution.
Also given differential equation does not have any singular solution.