Solving the RK4 design equations

I was digging into the Runge-Kutta method for solving differential equations and a line from [1] piqued my curiosity. These calculations, which are not reproduced in Kutta’s paper (they are however in Huen (1900)), are very tedious. The calculations are a set of eight constraints that the parameters of a fourth-order Runge-Kutta method must satisfy. I wondered how well Mathematica might have done at assisting Mr. Huen if it had been available in 1900. I go into Runge-Kutta methods in this post. Here I’d like to concentrate on a step in the design of the methods, namely solving the set of equations alluded in the quote above. The first thing to note is that there are 10 variables and only 8 equations, and so the solution is not fully determined. What we think of as the fourth order Runge-Kutta method is in fact a fourth order Runge-Kutta method. One could argue that we should have b2 = b3 and c2 = c3. With these additional equations, the system of equations has a unique solution, and…