In this paper a class of differential equations of the first order in the field of Mikusiński operators, $\mathcal{F}$, is considered. This class may correspond to a class of partial differential equations with constant coefficients. We construct a discrete analogue for these differential equations in $\mathcal{F}$, and obtain difference equations of the first order. We construct the approximate solutions of such difference equations and treat them as the approximate solution of the corresponding partial differential equation. Also we estimate the error of approximation.