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This thesis is concerned for presenting an efficient code for solving first and higher order non-stiff initial value problems (IVPs) of ordinary differential equation (ODEs). The proposed code will consist of block method based on Adams type formula. This developed block method manage to solve first and higher order ODEs problem directly in a single code using variable step size strategy. In order to gain an efficient numerical approximation, this developed block method will be implemented in the predictor corrector mode with simple iteration technique. Furthermore, this block method manage to compute several numerical solutions concurrently by depending the structure of the block. The coefficients of this developed block method formulae will be stored in the code in order to avoid the tedious and repetitive computation of the integration coefficients.
The stability properties of this developed block method is discussed and the local truncation error of the proposed scheme is studied. Several examples of IVPs of ODEs are taken into account to emphasize the main features of the proposed method and the results obtained show that the code manage to produce good results for solving first and higher order ODEs.