This paper studied the order and stability of a 2-Point Block Backward Difference method (2PBBD) for solving systems of nonstiff higher order Ordinary Differential Equations (ODEs) directly. The method computes the estimated solutions at two points concurrently within an equidistant block. The integration coefficients that are used in the method are calculated only once at the beginning of the programming. The numerical results obtained compare the stability region and error growth rate of the proposed method with 2-Point Block Divided Difference method (2PBDD) and 1-Point Backward Difference method (1PBD).

Nonlinear phenomena in science and engineering such as a periodically forced oscillator with nonlinear elasticity are often modeled by the Duffing oscillator (Duffing equation). The Duffling oscillator is a type of nonlinear higher order differential equation. In this research, a numerical approximation for solving the Duffing oscillator directly is introduced using a variable order stepsize (VOS) algorithm coupled with a backward difference formulation. By selecting the appropriate restrictions, the VOS algorithm provides a cost efficient computational code without affecting its accuracy. Numerical results have demonstrated the advantages of a variable order stepsize algorithm over conventional methods in terms of total steps and accuracy.