I'm trying to derive the error bound for the Ralston Runge-Kutta method (2nd order). Image 1 is taken from Ralston's book "A First Course in Numerical Analysis". In it he expands a step in y where h is the corresponding step in x using the Taylor series expansion. His purpose is to find Coefficients for the Runge Kutta method that produce the minimum error bound.
My problem is the operator D that he defines. What exactly is its function and why and how does it allow him to express the Taylor series in a different form?
In equation (5.6-10) it looks to me as if the RHS is a number and the LHS is still a function?
f(x,y) = y'(x) is the ODE he is examining.