 Apparent regular singular point

A regular singular point with a full set of analytic local solutions is an apparent singularity of the differential equation. Singular normalized coefficients do not by themselves force singular solutions. For example, $x{y}^{\prime \prime }-y^{\prime }=0$ has normalized coefficient $-1/x$, but its fundamental solutions $1,x^2$ are both analytic through zero. Their Wronskian is $2x$, which vanishes at the singular point while being nonzero on each adjacent ordinary interval. The nonvanishing-Wronskian criterion for a fundamental set applies on intervals where the normalized equation has regular coefficients.