Analytic approximations of solutions to systems of ordinary differential equations with variable coefficients

In this paper, reproducing kernel method is presented to obtain the analyticaland approximate solutions to system of ordinary differential equations with variablecoefficients. The solution obtained by using the method takes the form ofa convergent series with easily computable components. In the mean time, theapproximate solution un(x) is obtained by the n-term intercept of the analyticalsolution and is proved to converge to the exact solution. Some numerical examplesare studied to demonstrate the accuracy of the present method. Results obtainedby the method indicate the method is simple and effective.