We deal with algorithms for solving systems z′(x) = f(x, z(x)), x ε [0, c], z(0) = η where f has r continuous bounded derivatives in [0, c) × ^s. We consider algorithms whose sole dependence on f is through the values of n linear continuous functionals at f. We show that if these functionals are defined by partial derivatives off then, roughly speaking, the error of an algorithm (for a fixed f) cannot converge to zero faster than n^−r as n → +∞. This minimal error is achieved by the Taylor algorithm. If arbitrary linear continuous functionals are allowed, then the error cannot converge to zero faster than n^−(r+1) as n → +∞. This minimal error is achieved by the Taylor-integral algorithm which uses integrals of f.