The present paper provides the basis for the extension of the results of U. Ascher and the author [SIAM J. Numer. Anal. 20, 537-557 (1983; Zbl 0523.65064)] to more general problems. An analysis of the box and trapezoidal schemes applied to boundary value problems for linear singularly perturbed first order systems with variable coefficients (without turning points) is given. Stability and convergence results are derived for solution of the difference equation arising from the discretization of the mentioned problem. Some numerical results are presented.