  In this paper the necessary and sufficient conditions for a mapping to be the
dependence of the complete solution of some differentiable first-order ordinary
differential equation on the initial Cauchy condition are deduced. The result
is obtained by studying the Cauchy atlas on a manifold of complete solutions.
The proof is constructive - the corresponding differential equation is
obtained. The autonomous case and the linear case are discussed. The relation
to the Sincov functional equation is clarified.
