(a) Accurate starting is important in (7a), (7b). Illustrate this in Example 1 of the text by using starting values from the improved Euler?Cauchy method and compare the results with those in Table 21.8.

(b) How much does the error in Prob. 11 decrease if you use exact starting values (instead of RK values)?

Data from Prob. 11

Do and show the calculations leading to (4)?(7) in the text.

(c) Experiment to find out for what ODEs poor starting is very damaging and for what ODEs it is not.

(d) The classical RK method often gives the same accuracy with step 2h as Adams?Moulton with step h, so that the total number of function evaluations is the same in both cases. Illustrate this with Prob. 8. (Hence corresponding comparisons in the literature in favor of Adams?Moulton are not valid. See also Probs. 6 and 7.)

Data from Prob. 6

Solve the initial value problem by Adams?Moulton (7a), (7b), 10 steps with 1 correction per step. Solve exactly and compute the error. Use RK where no starting values are given

y' = (y - x - 1)² + 2, y(0) = 1, h = 0.1, 10 steps

X BEM h = 0.05 BEM h = 0.2 0.0 1.00000 0.1 0.26188 0.2 0.10484 0.3 0.10809 0.4 0.16640 0.5 0.25347 0.6 0.36274 0.37792 0.7 0.49256 0.8 0.64252 0.65158 0.9 0.81250 1.0 1.00250 1.01032 1.00000 0.24800 0.20960 Euler h = 0.05 Euler h = 0.1 0.15750 0.24750 0.35750 0.48750 0.63750 0.80750 0.99750 1.00000 - 1.00000 1.04000 RK h = 0.1 1.00000 1.00000 0.00750 0.34500 0.03750 0.15333 0.08750 -0.92000 0.12944 1.16000 0.17482 -0.76000 0.25660 1.36000 0.36387 -0.52000 0.49296 1.64000 0.64265 -0.20000 0.81255 2.00000 1.00252 3168 RK h = 0.2 1.000 5.093 25.48 127.0 634.0 Exact 1.00000 0.14534 0.05832 0.09248 0.16034 0.25004 0.36001 0.49001 0.64000 0.81000 1.00000