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I have found a toy implementation of RK-Dopri(4,5), written in Python.

I am concerned however, about line 118:

y = y + h * (b1*K1+b3*K3+b4*K4+b5*K5+b6*K6)

Has the term b2*K2 been omitted on purpose? I have a feeling that it is not, given (1.8) in Hairer, Wanner and Norsett:

enter image description here

Can you confirm?

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$b_2$ is zero in the Butcher tableau for Dormand-Prince, so you can skip that term.

Note that also the code you linked to contains lines b2 = 0.0, b2p = 0.0, which are just there as a reminder because those variables are never used.

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