Some problems of parametric integration of differential equations (Q1775200)

The authors present a qualitative study of the solutions of a first-order implicit differential equation using the method of parametrization. Under the hypothesis of a smooth differential equation, the authors rise the following questions: a) Will every solution of the parametrized equation generate a solution of the given differential equation? b) Which properties of solutions of the parametrized equation lead to an arbitrary ordinary solution (or to a singular solution) of the given differential equation? The paper tries to give answers to the questions a) and b). The particular case, when the differential equation is explicit, is also discussed.