On singular solutions of implicit second-order ordinary differential equations (Q1423461)

The author uses a geometric approach to study singular solutions of scalar implicit second-order ordinary differential equations \(F(x,y,y',y'')=0\) extending earlier works by Izumiya and Fukuda, resp., on the first-order case. The main results give necessary and sufficient conditions for the existence of a complete general or singular, resp., solution in the neighbourhood of a point in terms of the contact singular set under the assumption that \(0\) is a regular value of \(F_{y''}\) on the differential equation.