We give the generalizations of the classical theorems of Cauchy and Cauchy-Kovalevskaya on the existence and uniqueness of the analytical solution of a system of analytical and differential equations. The generalizations are given in terms of Power Geometry and describe situations when we can stop the computation of following terms of the local or asymptotic power expansion of a solution of the system of equations (§1). Further generalizations are also given (§2). We show how to apply the theory in the complicated cases; the system of algebraic equations and the autonomous ODE system are considered in detail (§3). As examples, we consider the Lorenz system (§4) and the Hill problem (§5).

Mathematical problems and theory of numerical methods