Aproximación de las soluciones de ecuaciones diferenciales de primer orden por medio de métodos numéricos

Abstract

This work explores the effectiveness of two numerical methods: Euler and Runge-Kutta, used to solve initial value problems. First, we analytically solve the differential equations and then apply both numerical methods to approximate their solutions. By comparing the approximations with the analytical solutions, we find that the Runge-Kutta method is more effective for the second initial value problem due to its lower error. Furthermore, we deduce that the smaller the interval size used for the approximation, the more reliable it will be compared to the analytical solution of the initial value problem. Subsequently, we applied the same procedure to a "simple" third initial value problem. The results showed that the previous assertion did not hold, revealing a high sensitivity to initial conditions that led to chaotic behavior. This finding aligns with chaos theory, indicating that small variations can trigger dramatically different outcomes. Finally, this study demonstrates that numerical methods are not always effective for all initial value problems, particularly the Runge-Kutta method for solutions that exhibit discontinuities, singularities, or high oscillation, leading to chaotic behaviors. This study underscores the importance of understanding the limitations of numerical methods in predicting solutions for initial value problems and paves the way for future work in improving these techniques and understanding chaos in dynamic systems.