Analysis of One Step Method for Solving Initial Value Problem of First Order Ordinary Differential Equations

ABSTRACT
The numerical solution of ordinary differential equations (ODEs), particularly
first-order initial value problems (IVPs), is essential in various scientific and
engineering applications where analytical solutions are difficult or impossible
to obtain. This study focuses on the analysis of one-step methods for solving
first-order initial value problems, emphasizing their mathematical
formulation, computational efficiency, and practical implementation. Onestep methods, such as Euler’s method and the Runge-Kutta methods, are
explored in detail, highlighting their advantages in approximating solutions
with high accuracy and stability. The research investigates the theoretical
properties of these methods, including consistency, convergence, and
stability, which are crucial in ensuring reliable numerical approximations. The
derived numerical scheme was implemented in MATLAB, where several test
cases were used to evaluate its performance. The numerical results were
compared with exact solutions to assess accuracy, error behavior, and
computational efficiency. The findings of this study demonstrate that one-step
methods provide a powerful and effective approach for solving IVPs in ODEs,
with applications spanning physics, engineering, finance, and biological
systems. Additionally, the research identifies challenges associated with stepsize selection and numerical stability, suggesting potential areas for further
improvements in adaptive and hybrid computational techniques. Ultimately,
this study contributes to the ongoing development of efficient and robust
numerical methods for solving differential equation