To show that the modified Euler method builds in the curvature term,
we start from the Taylor series expansion of (5) and
combine it with a backward Taylor series expansion about :

The difference between the second derivative at and at is of order . So if we evaluate the second derivative term in the expansion above at instead of , namely, , its contribution changes by an amount of net order , which may be absorbed in the last term. Solving this equation for then gives

Averaging this equation with (5) then gives

Notice that the second derivative term is gone. Finally, substituting the definitions of the approximate values and of the discrete time and using the differential equation (3) leads to (6).

Carleton DeTar 2008-12-01