Both are variable-step Runge-Kutta methods: RKF45 seems to be one of the default choices provided in MAPLE, and DOPRI5 is embedded in MATLAB’s universal solver ode45.
I don't know their performance and how to choose one, except the final apporach——using both solvers directly on the second-order ODE I want to solve and comparing the results.
P.S.
The paper "Solving Nonstiff Ordinary Differential Equations—The State of the Art" published in 1976, expresses a slight preference for RKF45, but its conclusions may be outdated.
Though DeepSeek recommends DOPRI5, I remain cautious about relying on it for specialized applications.
I have performed many comparisons of various methods. While there are some generalizations to be made, it does depend on the problem in question. I have attached one example of such a comparison. The bigger question is, "Why would anyone use MATLAB when you can solve such problems with a few lines of C?"
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