 Continuation through a zero of the leading ODE coefficient

An equation $a(x)y^{\prime \prime }+b(x)y^{\prime }+c(x)y=0$ 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.