As the demand for scientific and engineering computing continues to surge, ordinary differential equations (ODEs) have become indispensable in a wide array of fields, including physics, climate modeling, and artificial intelligence. However, when it comes to solving ODEs, traditional hardware built on the von Neumann architecture encounters significant bottlenecks in terms of both speed and energy efficiency. In response to these challenges, researchers are actively exploring innovative hardware architectures. Among them, memristors have emerged as a particularly promising option, thanks to their exceptional energy efficiency and robust parallel processing capabilities. While notable strides have been made in developing memristor-based solvers for partial differential equations, adapting these solutions directly to ODEs presents considerable difficulties. Achieving the necessary accuracy often requires reprogramming a vast number of devices or consuming substantial computational resources, both of which contribute to increased system complexity and hinder practical applications. Hence, there's an urgent need to devise efficient and precise methods for solving ODEs.