Page 218 - Schaum's Outline of Differential Equations

CHAP. 20] SOLVING SECOND-ORDER DIFFERENTIAL EQUATIONS 201

Continuing in this manner, but rounding to seven decimal places, we generate Table 20-3.

Table 20-3

Method: RUNGE-KUTTA METHOD
Problem: y"-y = x; y(0) = 0, /(O) = 1

ft = 0.1
x n
True solution
y n z n Y(x) = e*-e~ -x
x
0.0 0.0000000 1.0000000 0.0000000
0.1 1.1003333 1.0100083 0.1003335

0.2 0.2026717 1.0401335 0.2026720
0.3 0.3090401 1.0906769 0.3090406

0.4 0.4215040 1.1621445 0.4215047
0.5 0.5421897 1.2552516 0.5421906

0.6 0.6733060 1.3709300 0.6733072
0.7 0.8171660 1.5103373 0.8171674

0.8 0.9762103 1.6748689 0.9762120
0.9 1.1530314 1.8661714 1.1530335

1.0 1.3504000 2.0861595 1.3504024

20.8. UsetheRunge-Kuttamethodtosolve/'-3}; + 2}; = 0;XO) = -l,/(0) = Oontheinterval [0,1] withA = 0.1.
Using the results of Problem 20.2, we have/(jc, y, z) = I, g(x, y, z) = 3z — 2y, X Q = 0,y Q = —1, and z 0 = 0. Then,
using (20.4), we compute: