Singular Initial Value Problems for Some Quasi-Linear Second-Order Ordinary Differential Equations

We study existence, (non-)uniqueness and regularity of one- and two-sided solutions of singular initial value problems for second-order quasi-linear differential equations of the form \(g(u)u''=f(x,u,u')\) $g(u)u^{\prime \prime }=f(x,u,u^{\prime })$ with initial conditions \(u(y)=c_{0}\) $u(y)=c_0$ and \(u'(y)=c_{1}\) $u^{\prime }(y)=c_1$ where \(c_{0}\) $c_0$ is a simple zero of g and \(f(y,c_{0},c_{1})=0\) $f(y,c_0,c_1)=0$ . Our approach is based on the geometric theory of differential equations and in particular on singularity theory.