(a) Replace the so-called two-term Taylor estimate (equation (8) in Problem 23) by the three-term result:

Compute the second derivative y" in terms of f, and fy (partial derivatives of f, with respect to t and y, respectively), and deduce the three-term Taylor approximation:

(b) Show that the local discretization error en+l in this scheme is O(h3).
(c) Apply the method t o the I V P i n Problem 1 and compare the results.
(d) Repeat (c) for Problem 2.

y(, + h) = y(rn) + y'(fn)h