singular solution

Most solutions of a first-order equation belong to one tidy family with a free constant: pick a value of the constant, get a curve. A singular solution is the rebel — a genuine solution of the equation that you cannot get by ANY choice of that constant. It is an exception that lives outside the parameterized list, which is why 'general solution' and 'every solution' are not always the same thing.

Geometrically the most important singular solutions are envelopes: a curve that is tangent to every member of a one-parameter family, hugging the family as a smooth outer boundary. Because the singular solution touches each ordinary solution tangentially, the two share a point and a slope there, so the conditions of the existence-and-uniqueness theorem fail at those touching points — uniqueness breaks, and a second solution branches off. That is the deep reason singular solutions can exist at all: they appear exactly where uniqueness is not guaranteed.

You meet them most cleanly in Clairaut's equation, whose line family has a curved envelope, but they also lurk wherever you divided by something that can be zero. Separating variables, for instance, can throw away a constant equilibrium solution that turns out to be singular. The honest takeaway: after finding the general solution, pause and ask whether the equation has extra solutions hiding outside the family — the mathematics is not always as complete as the tidy formula suggests.