an irregular singular point
An irregular singular point is the hard case: a singular point where the coefficients blow up too violently for even the method of Frobenius to be guaranteed. It is the place where the well-stocked toolbox of series solutions runs out of guarantees, and the solution's behaviour near the point can be wildly more complicated than any power times a tidy series.
The classification is just the failure of the regular-singular test. At a singular point x0 of y'' + P(x) y' + Q(x) y = 0, you check whether (x - x0) P(x) and (x - x0)^2 Q(x) are both analytic. If at least one of them is still not analytic — P blows up faster than 1/(x - x0), or Q faster than 1/(x - x0)^2 — then x0 is an irregular singular point. A standard example is y'' + (1/x^2) y' + y = 0 at x = 0: here x P = 1/x still blows up, so the regular test fails and x = 0 is irregular.
Why does it matter to name this case? Because near an irregular singular point solutions can carry essential singularities — think of something like e^(1/x), which has no Taylor or Frobenius expansion at all — so the elementary series machinery may produce no usable solution. This is a genuine boundary of the method: at irregular singular points one turns to asymptotic expansions, special transforms, or numerical work. It is an honest reminder that the power-series program, powerful as it is, does not solve everything.
For x^3 y'' + y = 0 at x = 0, standard form has Q = 1/x^3, so x^2 Q = 1/x still blows up at x = 0. The regular-singular test fails, so x = 0 is an irregular singular point and a Frobenius series is not assured.
Just one failed condition — here x^2 Q still singular — is enough to brand the point irregular.
Irregular does not mean 'no solution exists' — solutions always exist — only that the standard series methods lose their guarantee; you may still find solutions by asymptotic or numerical means.