Penseive header: checking the local error in Runge-Kutta.

eqn= y'[x]-f[x,y[x]]

Series[eqn, {x,0,1}]

(#==0)& /@ CoefficientList[Series[eqn, {x,0,1}], x]  /. y[0] -> 0

n=3;
Solve[
(#==0)& /@ CoefficientList[Series[eqn, {x,0,n}], x]  /. y[0] -> 0,
Table[Derivative[k][y][0], {k, 1, n+1}]
]

(Subscript[k, 1]) = h f[0,0];
(Subscript[k, 2]) = h f[(h/2), (1/2)(Subscript[k, 1])];
(Subscript[k, 3]) = h f[(h/2), (1/2)(Subscript[k, 2])];
(Subscript[k, 4]) = h f[h, (Subscript[k, 3])];
y = (1/6)((Subscript[k, 1])+2(Subscript[k, 2])+2(Subscript[k, 3])+(Subscript[k, 4]))

Series[y, {h,0,4}]

f[0,0] ((Derivative[0,1][f][0,0])^3)+4 ((f[0,0])^2) Derivative[0,1][f][0,0] Derivative[0,2][f][0,0]+((f[0,0])^3) Derivative[0,3][f][0,0]+((Derivative[0,1][f][0,0])^2) Derivative[1,0][f][0,0]+3 f[0,0] Derivative[0,2][f][0,0] Derivative[1,0][f][0,0]+5 f[0,0] Derivative[0,1][f][0,0] Derivative[1,1][f][0,0]+3 Derivative[1,0][f][0,0] Derivative[1,1][f][0,0]+3 ((f[0,0])^2) Derivative[1,2][f][0,0]+Derivative[0,1][f][0,0] Derivative[2,0][f][0,0]+3 f[0,0] Derivative[2,1][f][0,0]+Derivative[3,0][f][0,0] == f[0,0] ((Derivative[0,1][f][0,0])^3)+4 ((f[0,0])^2) Derivative[0,1][f][0,0] Derivative[0,2][f][0,0]+((f[0,0])^3) Derivative[0,3][f][0,0]+((Derivative[0,1][f][0,0])^2) Derivative[1,0][f][0,0]+3 f[0,0] Derivative[0,2][f][0,0] Derivative[1,0][f][0,0]+5 f[0,0] Derivative[0,1][f][0,0] Derivative[1,1][f][0,0]+3 Derivative[1,0][f][0,0] Derivative[1,1][f][0,0]+3 ((f[0,0])^2) Derivative[1,2][f][0,0]+Derivative[0,1][f][0,0] Derivative[2,0][f][0,0]+3 f[0,0] Derivative[2,1][f][0,0]+Derivative[3,0][f][0,0]