The system of linear partial differential equations (PDE) of the first order, with initial conditions given by the HEAVISIDE (step) function, is considered. It is shown that the system can be solved analytically (in the terms of integrals), where the problem of numerical quadratures arises. Some integrals are improper (singular in the upper bound), but considering their convergence and calculating their principal value (in a CAUCHY sense), the problem can be successfully solved. In the special case the achieved results have been significantly simplified and then compared with the numerical solution obtain-ed by the application of the finite differences method, namely the up-wind scheme, which is used most often for that kind of problems. A detailed analysis of the obtained error, with some graphical illustrations for various values of parameters, is given. The proposed initial value problem has several applications, e.g., in semiconductor physics, where it is a good model for the inter-valley transfer in a two-valley semiconductor electron devices in the case when the electric field is stationary and homogeneous.