Rezidue method for solving ordinary differential equations

In this thesis the algebraic structure of the boundary value problem for linear differential equations in general form is comprehensively investigated. The properties of linear differential operators generated by the problem have also been studied. For this aim, at first the adjoint differential expression in Lagrange?s sense is written and the adjoint boundary conditions are examined. Moreover, the eigenvalues and eigenfunctions of the linear differential operators and of some properties have been studied. Later, the Green?s function of the boundary value problem for a linear differential equations has been obtained. By using the Green?s function, the concept of the inverse operator has been given, too. The results of the above investigated subjects are used for both solving the boundary value problem of linear differential equations when Fourier?s method is applied and verifying the obtained solution. It is known that if the eigenvalues are the higher order poles of the solution, corresponding eigenfunctions are not orthonormal and the problem of expansion of any function over these eigenfunctions is open. In this case, the application of Fourier?s method is impossible instead, the residue method for the boundary value problem of a linear differential equation with constant coefficients has been investigated. The residue method can be applied when the differential operator generated by the corresponding problem is not selfadjoint. Finally, using the residue method, some boundary value problems of the second order linear differential equations with constant coefficients have been solved.