Optimizing an implementation of the RKF method
All these lines are really weird:
a4  =   9.230769230769231e-01  #  12/13

Unless you have a good reason (which I'd then state in the code as a comment) to do that, just write a4 = 12/13 instead.
Gonna be the same anyway:
&gt;&gt;&gt; import dis
&gt;&gt;&gt; dis.dis('a4 = 12/13')
  1           0 LOAD_CONST               0 (0.9230769230769231)
              2 STORE_NAME               0 (a4)
              4 LOAD_CONST               1 (None)
              6 RETURN_VALUE
&gt;&gt;&gt; dis.dis('a4 = 9.230769230769231e-01')
  1           0 LOAD_CONST               0 (0.9230769230769231)
              2 STORE_NAME               0 (a4)
              4 LOAD_CONST               1 (None)
              6 RETURN_VALUE

This line for example is not right:
b51 =   2.032407407407407e+00  #  439/216

The values differ slightly, your value being less accurate:
&gt;&gt;&gt; 2.032407407407407e+00
2.032407407407407
&gt;&gt;&gt; 439/216
2.0324074074074074