Approximate solution of linear ordinary differential equations with variable coefficients

In this paper, a novel, simple yet efficient method is proposed to approximately solve linear ordinary differential equations (ODEs). Emphasis is put on second-order linear ODEs with variable coefficients. First, the ODE to be solved is transformed to either a Volterra integral equation or a Fredholm integral equation, depending on whether initial or boundary conditions are given. Then using Taylor’s expansion, two different approaches based on differentiation and integration methods are employed to reduce the resulting integral equations to a system of linear equations for the unknown and its derivatives. By solving the system, approximate solutions are determined. Moreover, an n th-order approximation is exact for a polynomial of degree less than or equal to n. This method can readily be implemented by symbolic computation. Illustrative examples are given to demonstrate the efficiency and high accuracy of the proposed method.