Numerical solution of the problem of a crossing point

In this paper we consider the problem of finding the point of intersection of the trajectories of two objects, the motion of one of which is described by a system of ordinary differential equations and is controlled by the initial conditions. This problem reduces to a boundary value problem in the theory of ordinary differential equations, which can be reduced to solving a system of nonlinear algebraic equations of a special kind. Discussed in this paper tasks have a distinctive feature. The right part of a system of ordinary differential equations can depend on initial conditions. We produce adaptation of Newton\'s method for solving arisen systems of nonlinear equations, analysis of heuristic false position method, his improvement and a comparison with Newton\'s method. In conclusion, the author analyzes the accuracy of the solution of the crossing point problem for the case of noisy input data.