On the Differential Equation of First and Second Order in the Zeon Algebra

In this paper, the study of linear differential equations involving one conventional and two nilpotent variables is started. This is a natural extension of the case of one involved nilpotent (para-Grassmann) variable studied earlier. In the case considered here, the two nilpotent variables are assumed to commute, hence they are generators of a (generalized) zeon algebra. Using the natural para-supercovariant derivatives D_i transferred from the study of a para-Grassmann variable, we consider linear differential equations of order at most two in D_i and discuss the structure of their solutions. For this, convenient graphical representations in terms of simple graphs are introduced.