Exact schemes for second-order linear differential equations in self-adjoint cases

When working with mathematical models, to keep the model errors as small as possible, a special system of linear equations is constructed whose solution vector yields accurate discretized values for the exact solution of the second-order linear inhomogeneous ordinary differential equation (ODE). This case involves a 1D spatial variable x with an arbitrary coefficient function κ(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\kappa (x)$\end{document} and an arbitrary source function f(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f(x)$\end{document} at each grid point under Dirichlet or/and Neumann boundary conditions. This novel exact scheme is developed considering the recurrence relations between the variables. Consequently, this scheme is similar to those obtained using the finite difference, finite element, or finite volume methods; however, the proposed scheme provides the exact solution without any error. In particular, the adequate test functions that provide accurate values for the solution of the ODE at arbitrarily located grid points are determined, thereby eliminating the errors originating from discretization and numerical approximation.