Riccati differential equations are one of the most common type of non-linear differential equation used to model real life applications from various fields. The issue when dealing with non-linear differential equations is obtaining their exact solutions. In this research, a three-point block multi-step method in backward difference form is introduced to provide approximated solutions for these Riccati differential equations. The accuracy of the proposed three-point block method will be tested against known numerical methods. The efficiency of the method will apparent when compared with another multi-step method.

Existing variable order step size numerical techniques for solving a system of higher-order ordinary differential equations (ODEs) requires direct calculating the integration coefficients at each step change. In this study, a variable order step size is presented for direct solving higher-order orbital equations. The proposed algorithm calculates the integration coefficients only once at the beginning and, if necessary, once at the end. The accuracy of the numerical approximation is validated with well-known orbital differential equations. To reduce computational costs, we obtain the relationship for the predictor-corrector algorithm between integration coefficients of various orders. The efficiency of the proposed method is substantiated by the graphical representation of accuracy at the total evaluation steps.