The Schwinger variational linear algebraic equations (VLA) method is generalised to deal with the case of coupled integro-differential equations for nonlocal as well as local interaction potentials and for arbitrary partial waves. Exploratory calculations employing the VLA method for the scattering of electrons by hydrogen atoms using the single-channel static-exchange and exchange-adiabatic approximations and the two channel 1s-2s close-coupling approximation, as well as for a schematic model involving square-well potentials, are discussed. Lower bounds to the phaseshifts are calculated in suitable cases.