The Runge-Kutta-Verner method (see [Ve]) is based on the formulas
wi+1 = wi + 13/160 k1 + 2375/5984 k3 + 5/16 k4 + 12/85 k5 + 3/44 k6 and
i+1 = wi + 3/40 k + 875/2244 k3 + 23/72 k4 + 264/1955 k5 + 125/11592 k7 + 43/616 k8,
Where
k1 = hf (ti ,wi),
k2 = hf(ti + h/6 ,wi + 1/6 k1),
k3 = hf(ti + 4h/15 ,wi + 4/75 k1 + 16/75 k2),
k4 = hf(ti + 2h/3 ,wi + 5/6 k1 - 8/3 k2 + 5/2 k3),
k5 = hf(ti + 5h/6, wi - 165/64 k1 + 55/6 k2 - 425/64 k3 + 85/96 k4),
k6 = hf(ti + h, wi + 12/5 k1 − 8k2 + 4015/612 k3 - 11/36 k4 + 88/255 k5),
k7 = hf (ti + h/15,wi - 8263/15000 k1 + 124/75 k2 - 643/680 k3 - 81/250 k4 + 2484/10625 k5),
k8 = hf (ti + h, wi + 3501/1720 k1 - 300/43 k2 + 297275/52632 k3 - 319/2322 k4 + 24068/84065 k5 + 3850/26703 k7).
The sixth-order method i+1 is used to estimate the error in the fifth-order method wi+1. Construct an algorithm similar to the Runge-Kutta-Fehlberg Algorithm, and repeat Exercise 3 using this new method.
In Exercise 3