We report elastic differential cross sections (DCS) for the scattering of electron by H2, CH4, C2H4, SiH4, and H2O in the 5 to 20 eV energy range. These cross sections were obtained using the Schwinger variational principle. In our procedures the scattering wave function is expanded as a combination of plane waves. In the present implementation the exchange potential and polarization effects are not considered and all results are obtained at the static level of approximation. For all molecules, except H2, our calculated DCS agree reasonably well with avaliable experimental data and with other theoretical results (obtained at the static-exchange level of approximation). Our study suggests that the Schwinger variational principle with plane wave expansion may become an efficient method for studying electron-molecule scattering.