A comparative study of first order differential equation by using Butcher’s fifth order method and Dormand Prince method

Nonlinear differential equations present considerable challenges when tackled with fundamental approaches due to their inherent complexity. Numerical methods like Euler, Taylor, and Runge-Kutta provide effective alternatives for solving these equations. The Butcher’s fifth-order method and Dormand Prince method are part of the Runge-Kutta family that have been used to address higher-order differential equations and have been applied in science and mathematics fields. Both methods are known for their high accuracy in solving such problems, as they can produce better approximation to the exact solutions. However, there are limited studies on solving the first-order Bernoulli and Riccati differential equations using these methods. Therefore, this study attempts to solve both differential equations using the Butcher’s fifth-order and Dormand Prince methods with a fixed step size of h = 0.1. Both methods were analysed by comparing their absolute errors against the exact solution. The results showed that both Butcher’s and Dormand Prince methods are capable of solving first-order Bernoulli and Riccati differential equations. However, in terms of accuracy, the findings highlight that the Butcher’s fifth-order method is better at solving problems 1 and 3, while the Dormand Prince method is suitable for solving problems 2 and 4, demonstrating more accurate approximations due to smaller errors. It proved that both methods are accurate and suitable for solving these problems. However, the accuracy of both methods depends on the specific problems. Future research could investigate their application to higherorder nonlinear differential equations with different step sizes. The accuracy and efficiency of both methods could be further evaluated by comparing relative errors, computational work, or execution time.