Runge-Kutta Methods with Constrained Minimum Error Bounds

Optimum Runge-Kutta methods of orders $m = 2,3$, and $4$ are developed for the differential equation $y’ = f(x,y)$ under Lotkin’s conditions on the bounds for $f$ and its partial derivatives, and with the constraint that the coefficient of ${\partial ^m}f/\partial {x^m}$ in the leading error term be zero. The methods then attain higher order when it happens that $f$ is independent of $y$.