This study investigates the theoretical and practical aspects of solving ordinary differential equations (ODEs) by employing integrating factors. In this work, we present an integrating factor approach for solving higher order differential equations and numerical and neural networks solutions. The findings demonstrate that through a continuous function as an integrating factor, one can elevate the order of any ODE, providing an efficient framework for analyzing complex differential systems. Key propositions illustrate how integrating factors contribute to transforming higher order ODEs, enabling systematic solutions. Both analytical and numerical approaches are discussed, including applications such as the Korteweg-de Vries (KdV) equation, illustrating the feasibility and reliability of the methods. Furthermore, neural networks are explored as a computational approach to approximate ODE solutions, providing insight into their efficacy and limitations in capturing oscillatory behaviors in differential systems.