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In this paper, we exploit an numerical method for solving second order differential equations with boundary conditions. Based on the theory of the analytic solution, a series of spline functions are presented to find approximate solutions, and one of them is selected to approximate the solution automatically. Compared with the other methods, we only need to solve a tri-diagonal system, which is much easier to implement. This method has the advantages of high precision and less computational cost. The analysis of local truncation error is also discussed in this paper. At the end, some numerical examples are given to illustrate the effectiveness of the proposed method.

A Class of Spline Functions for Solving 2-Order Linear Differential Equations with Boundary Conditions