To get singular solutions, do we always need a guess or experiment?
Can we get it from a relation of family of curves of general solution? For example, $(y')^2-xy'+y=0$ has the general solution $y=cx-c^2$. It has a singular solution of $y=x^2/4$, too. If you draw family of curves of general solution (a bunch of straight lines) as well as curve of singular solution (a parabola), you can find parabola is touching general family of curves with a pattern. Can that be a point to get singular solution?
In general, is there a way to calculate singular solutions mathematically?