= Continuation through a zero of the leading ODE coefficient
An equation $a(x)y''+b(x)y'+c(x)y=0$ can remain meaningful at a zero of $a$ even though its normalized <second-order linear differential equation> has singular coefficients. A global continuation problem must specify its solution regularity. Across a point where a homogeneous solution vanishes to second order, its coefficient may change in a piecewise $C^1$ continuation; requiring continuity of the second derivative can eliminate that freedom. The usual continuous-coefficient <uniqueness theorem for ordinary differential equations> cannot simply be applied across such a point.
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