The main purpose of this work is divided into four categories:- First, give a theoretical study of the linear double eigenvalue problems with an extension to the study of the linear multiparameter eigenvalue problems. Second, some methods, namely variational method, finite-difference method, collocation and Galerkin's methods are presented to solve the double linear eigenvalue problems related to the Sturm-Liouville ordinary differential equations. Third, we shed light on the methods of solution for the double linear eigenvalue problems associated with the integral and integro-differential equations. Fourth, some modification is devoted to give a new method to solve the non-eigenvalue problems related to the integral and integro-differential equation namely, the eigenfunction expansion method.