A recent posting of Mario Lemelin showed that Maple's default numerical methods produced wrong results for a certain differential equation. Further investigation revealed that the problem stemmed from the fact that the fourth and fifth order Runge-Kutta methods used within the rkf45 method both produce the same (exactly correct) result at any step size, causing the adaptive error analysis to go badly wrong. This leads to the question: when do Runge-Kutta methods produce exact results for arbitrary step sizes and initial conditions?
For a partial answer, see this worksheet: View 4541_runge.mw on MapleNet or Download 4541_runge.mw
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