(untitled) (CAHIERS MAGELLANES-NS, 2024)
First-order linear or nonlinear ordinary differential equations (ODEs) can be solved with the help of single-step or multistep numerical methods. This paper discusses multistep numerical methods. With the goal of producing a more efficient multistep numerical method, this work will construct a general implicit method (GI3SM) with a third derivative to directly solve the general category of quasi-linear first-order Ordinary Differential Equations (ODEs), which is expressed as . The necessary Hermite interpolating polynomials with (GI3SM) have been derived in three steps (IVPs) in order to implement the new efficient multistep numerical technique. A multi-step method is developed for solving this problem, provided that the numerical approximation at three steps is acceptable. To complete the derivation of the multistep implicit method for solving ordinary differential equations of the first order by adding a third derivative. To evaluate the effectiveness of the process, we have looked at four tested examples. The accuracy and efficacy of the suggested method are contrasted to Classical RK and Euler numerical methods with the exact solutions to these problems using the numerical solutions of the implementations. In addition to studying the order and zero-stability of the proposed method, a number that of characteristics have also been established. The implicit multistep proposed method (GI3SM) yields result that are in good agreement with analytical solutions when compared to the classical RK and Euler methods. Additionally, the (GI3SM) generates exact numerical answers for the test problems.