In sophomore and junior level ordinary differential equations one studies the classical Sturm-Liouville boundary value problem, where the boundary conditions are of the separated type. It is well lmown that under very reasonable hypotheses this problem has a discrete set of non- trivial solutions f o r a discrete set of eigenvalues which are countably infinite and tend to infinity. It is the purpose of this thesis to study the question of whether similar results hold for problems when the boundary conditions are replaced by conditions of the non-separated type and also conditions where an integral is added. - In doing so, we are able to generalize some recent results of Etgen and Tefteller.