Various real processes, occurring in the nature, technology, etc., are usually described by differential equations. Due to the development of computer software, computers have become the main tool for solving problems of different fields. They enable not only to solve complex differential equations or their systems, but also to analyze the dependence of differential equations solutions on various parameters and initial values. Up to the present many methods for the solution of differential equations have been developed, therefore, the user can solve differential equation, using several different methods. Different methods of solution enable to avoid various mistakes and to reduce errors. Differential equations can be solved not only using numerical methods, but also by applying methods of algebraic operator equations. When the latter method is being used, solutions are expressed in power series, the convergence of which has to be analyzed separately. This paper includes the analysis of Mathieu-type differential equations solutions dependence on initial conditions and parameters, as well as the establishment of solutions attractor zones and curves, which separate different attractor zones. It is very important to indicate the most exact crossing limits among different attractor zones. In order to avoid huge errors, we carried out the research by using two methods: operator and Runge-Kutta.