A non-iterative derivative-free method for nonlinear ordinary differential equations

Most methods for the determination of approximate solutions of nonlinear ordinary differential equations require that the nonlinearities be sufficiently differentiable with respect to the dependent variable and its derivatives. In this paper, a non-iterative method for obtaining approximate solutions of nonlinear ordinary differential equations which does not require the derivatives of the nonlinearities is presented and its convergence is proved. The method is also generalized for the determination of the periodic solutions of autonomous, nonlinear ordinary differential equations by introducing as dependent variable the unknown frequency of oscillation. The method does not require the presence of small parameters and can be used in a piecewise fashion.