Classes of finite order formal solutions of an ordinary differential equation

In this paper we select general classes of finite order formal solutions of an algebraic (polynomial) ordinary differential equation (ODE), that can be calculated by the methods of the planar power geometry based on determining leading terms of the equation by the Newton–Bruno polygon.
Beside that in this paper we prove the theorem that if a formal solution of the selected class exists then the first approximation (the truncation) of this solution is a (formal) solution of the first approximation of the initial equation (that is called the truncated equation). Calculated formal solutions by means of these methods relate to much more general classes of the formal solutions that are called grid-based series and transseries in the foreign papers. Grid-based series and transseries are fairly new objects and in spite of the large number of publications they are slightly studied. Such series appear among formal solutions of differential equations including equations that are important in physics. Other general methods of the calculation of such series do not exist yet. Therefore it is important to select the classes of formal solutions that can be calculated algorithmically by the methods of the planar power geometry.
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