Page 385 - Schaum's Outline of Differential Equations

368 ANSWERS TO SUPPLEMENTARY PROBLEMS

20.24. Method: ADAMS-BASHFORTH-MOULTON METHOD

Problem: /' + y = 0; y(0) = 1 , /(O) = 0
ft = 0.1
x n
True solution
py n PZ n y n •^n Y(x) = cos x
•7
0.0 — — 1.0000000 0.0000000 1.0000000

0.1 — — 0.9950042 -0.0998333 0.9950042
0.2 — — 0.9800666 -0.1986692 0.9800666
0.3 — — 0.9553365 -0.2955200 0.9553365

0.4 0.9210617 -0.3894147 0.9210611 -0.3894184 0.9210610

0.5 0.8775837 -0.4794223 0.8775827 -0.4794259 0.8775826
0.6 0.8253371 -0.5646396 0.8253357 -0.5646431 0.8253356
0.7 0.7648439 -0.6442153 0.7648422 -0.6442186 0.7648422

0.8 0.6967086 -0.7173541 0.6967066 -0.7173573 0.6967067

0.9 0.6216119 -0.7833254 0.6216096 -0.7833284 0.6216100
1.0 0.5403043 -0.8414700 0.5403017 -0.8414727 0.5403023

20.25. Since the true solution is Y(x) = —x, a first-degree polynomial, the Adams-Bashforth-Moulton method is exact and
generates the true solution y n=-x n at each x n.

20.26. Method: MILNE'S METHOD
Problem: y" -3y' + 2y = 0; y(0) = -1 , /(O) = 0

h = 0.l
x n
True solution
py n PZ n y n •^n Y(x) = e - 2e x
2x
•7
0.0 — — -1.0000000 0.0000000 -1.0000000
0.1 — — -0.9889417 0.2324583 -0.9889391

0.2 — — -0.9509872 0.5408308 -0.9509808
0.3 — — -0.8776105 0.9444959 -0.8775988

0.4 -0.7582563 1.4671290 -0.7581224 1.4674042 -0.7581085
0.5 -0.5793451 2.1387436 -0.5791820 2.1390779 -0.5791607

0.6 -0.3243547 2.9955182 -0.3241479 2.9959412 -0.3241207
0.7 0.0274045 4.0823034 0.0276562 4.0828171 0.0276946

0.8 0.5015908 5.4542513 0.5019008 5.4548828 0.5019506
0.9 1.1299955 7.1791838 1.1303739 7.1799534 1.1304412

1.0 1.9519398 9.3404286 1.9524049 9.3413729 1.9524924