  In this paper, the necessary and sufficient conditions in order that a smooth
mapping F be a dependence of a complete solution of some second-order ordinary
differential equation on Neumann conditions are deduced. These necessary and
sufficient conditions consist of functional equations for F and of a smooth
extensibility condition. Illustrative examples are presented to demonstrate
this result. In these examples, the mentioned functional equations for F are
related to the functional equations for geodesics, to Jensen's equation, to the
functional equations for conic sections and to Neuman's result for linear
ordinary differential equations.
