Page 226 - Schaum's Outline of Differential Equations

CHAP. 20] SOLVING SECOND-ORDER DIFFERENTIAL EQUATIONS 209

Continuing in this manner, we generate Table 20-8.

Table 20-8

Method: MILNE'S METHOD

Problem: f-y = x; y(0) = 0, /(O) = 1
h = 0.l
x n
True solution
py n PZn y n z n Y(x) = <?-e~ -x
x
0.0 — — 0.0000000 1.0000000 0.0000000

0.1 — — 0.1003333 1.0100083 0.1003335

0.2 — — 0.2026717 1.0401335 0.2026720
0.3 — — 0.3090401 1.0906769 0.3090406
0.4 0.4214983 1.1621433 0.4215045 1.1621445 0.4215047

0.5 0.5421838 1.2552500 0.5421903 1.2552517 0.5421906

0.6 0.6733000 1.3709276 0.6733071 1.3709300 0.6733072
0.7 0.8171597 1.5103347 0.8171671 1.5103376 0.8171674
0.8 0.9762043 1.6748655 0.9762120 1.6748693 0.9762120

0.9 1.1530250 1.8661678 1.1530332 1.8661723 1.1530335
1.0 1.3503938 2.0861552 1.3504024 2.0861606 1.3504024

Supplementary Problems

20.15. Reduce the initial-value problem y" + y = 0; y(G) = 1, /(O) = 0 to system (20.1).
20.16. Reduce the initial-value problem y"-y = x; y(Q) = 0, /(O) = -1 to system (20.1).