ordinary and singular points

Before you try to solve a differential equation with a power series, you want to know which points are friendly and which are treacherous. For a second-order linear equation, an ordinary point is a place where the equation is well behaved and you can confidently expand the solution in a Taylor series; a singular point is a place where something in the equation blows up, and the naive series may fail. It is a bit like asking, before driving, whether the road ahead is smooth pavement or a pothole.

Write the equation in the standard form y'' + P(x) y' + Q(x) y = 0, with the leading coefficient divided out. A point x0 is an ordinary point if both P(x) and Q(x) are analytic there, meaning each has a convergent Taylor expansion around x0. If either P or Q fails to be analytic at x0 — typically because the original leading coefficient vanished there and you divided by zero — then x0 is a singular point. For example, for x^2 y'' + x y' + (x^2 - n^2) y = 0 (Bessel's equation) you divide by x^2 to get P(x) = 1/x and Q(x) = 1 - n^2/x^2, both of which blow up at x = 0, so x = 0 is a singular point; every other point is ordinary.

The distinction is not bureaucratic: it decides which tool you reach for. At an ordinary point an ordinary power series y = sum a_k (x - x0)^k works and gives two independent analytic solutions. At a singular point you generally cannot just plug in a Taylor series — the solution may have a fractional power, a logarithm, or a genuine blow-up — so you need the more careful method of Frobenius, and only if the singularity is mild enough (a regular singular point). The named equations of physics — Legendre, Bessel, Hermite — are precisely interesting because their singular points sit exactly where the physics lives (the poles of a sphere, the axis of a cylinder).