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 \(\kappa (x)\) and an arbitrary source function \(f(x)\) 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.

在使用数学模型时，为了使模型误差尽可能小，构建了一个特殊的线性方程组系统，其求解矢量可为二阶线性非齐次常微分方程（ODE）的精确解提供精确的离散值。这种情况涉及一维空间变量x，它具有任意系数函数\（\ kappa（x）\）和任意源函数\（f（x）\）在Dirichlet或/和Neumann边界条件下的每个网格点处。考虑变量之间的递归关系，开发了这种新颖的精确方案。因此，该方案类似于使用有限差分法，有限元法或有限体积法获得的方案。但是，提出的方案提供了准确的解决方案，没有任何错误。特别是，确定了为在任意位置的网格点处的ODE解决方案提供准确值的适当测试功能，从而消除了因离散化和数值近似而产生的误差。