Singular Points

Differential Equations · Axiom Academy

Every point on the line is either "ordinary" (business as usual) or "singular" (something breaks). Learn to spot the difference just by looking at the equation. A point where the equation breaks Take the differential equation xy'' + y' - y = 0 . It looks perfectly ordinary — until you try to solve it near x = 0 . Watch what happens to its coefficient functions as a test point sweeps across the number line. Every point on the line gets tested. Watch the live P(x) and Q(x) readout as the marker sweeps — it stays calm and green almost everywhere, then spikes red exactly once. The equation in standard form is — and is undefined at exactly one place. Pick a point — is it ordinary or singular? Same equation, xy'' + y' - y = 0 . Click each point below and watch the coefficients P(x) and Q(x) get evaluated live. One of these seven points is going to misbehave. x = 0 is the only place where the leading coefficient a_2(x) = x vanishes — and that's exactly where P and Q blow up. A trickier equation — two singular points this time Now try (x-3)x^2y'' + 2xy' - y = 0 . Click each point below and classify it as ordinary or singular . Think about where the leading coefficient a_2(x) = (x-3)x^2 vanishes before you click. a_2(x) = (x-3)x^2 = 0 at x = 0 and x = 3 — exactly the two singular points, no others. The general rule — and what's next You've just discovered how to spot a singular point using nothing but the coefficient functions. Here's the rule in full:

This is the written version of the interactive lesson above. See the full Differential Equations course.