@nm

Check **?PDEtools:-casesplit** and the glossary to understand what is Maple doing to compute singular solutions. In brief, as in the title, and I suggest the DifferentialThomas engine, if you see more than the general case then expect a solution by dsolve for that extra case, whenever that extra solution is computable in practice.

The documentation for **dsolve**, also, explains clearly that by default it returns all *essential* singular solutions. If you don't want them, then use *singsol = false*. If you want also the *non-essential* ones, provided they exist (they appear as additional cases when passing the ODE (possibly system) to PDEtools:-casesplit), *singsol = all* will make them be returned as well. All this is explained in ?dsolve,details if I remember right.

Regarding your several "let's think that this implies that, which perhaps implies something else, and why not this other thing," I cannot keep responding to every single conjecture that you imagine might be the case. Lie methods are what they are - explained in the documentation. When in action, singular cases that are not *essential* may pop up. However, if you let dsolve do its job without trying to guide it, only the *essential* singular cases are considered. That is all.

For your **ode := diff(y(x), x) = (2*x)*sqrt(1 - y(x)^2),** the limit when **_C1=infinity** mentioned in the previous reply already answers that one too. Regarding your "I have dozens of examples of this and that and perhaps something else", I cannot answer that. Regarding your solutions y = r and y = -r for some ODE that you do not show, I also cannot answer that.

I realize you want answers to so many questions that pass through your mind, but I don't think this is the forum for all and every single conjecture that you could imagine. Anyway ...

Best

**Update** Re-reading the title of your question, what is your point with asking in negatives? Clearly, **dsolve** can find, and indeed finds singular solutions, with the luxury of *all, essential* or *none.* If however I were a bit sleepy when reading the title of your question, or would skim quickly over the contents, I'd think otherwise. It would help for answering your questions if the title is not just your conjecture of negatives.