Singular Initial Value Problems for Scalar Quasi-Linear Ordinary Differential Equations

We discuss existence, non-uniqueness and regularity of one- and two-sided solutions of initial value problems for scalar quasi-linear ordinary differential equations where the initial condition corresponds to an impasse point of the equation. With a differential geometric approach, we reduce the problem to questions in dynamical systems theory. As an application, we discuss in detail second-order equations of the form $g(x)u^{\mathrm{\prime }\mathrm{\prime }}=f(x,u,u^{\mathrm{\prime }})$ with an initial condition imposed at a simple zero of $g$. This generalises results by Liang and also makes them more transparent via our geometric approach.