Solutions of the Fifth-Order Runge-Kutta Equations

(a) Under the restriction R₂ = 0, the sixteen equations in twenty-one unknowns which specify a fifth-order Runge-Kutta formula are so arranged as to give a solution which is completely rational in terms of five parameters. (b) If R₂ ≠ 0, the solutions fall into two classes. The simpler class permits a solution involving a single quadratic irrationality. The other class is reduced to the simultaneous solution of two determinantal equations each involving two unknowns and five parameters.