Chapter 9 - In Review1.$x_n$Euler $h=0.1$Euler $h=0.05$Imp.
Euler$h=0.1$Imp. Euler$h=0.05$RK4$h=0.1$RK4$h=0.05$
1.001.051.101.151.201.251.301.351.401.451.502.0000
2.1386
2.3097
2.5136
2.7504
3.0201 2.00002.06932.1469 2.2328 2.32722.4299 2.54092.6604
2.78832.9245 3.06902.0000
2.1549
2.3439
2.5672
2.8246
3.11572.00002.07352.15542.24592.34502.45272.56892.69372.82692.96863.11872.0000
2.1556
2.3454
2.5695
2.8278
3.11972.00002.07362.15562.24622.34542.4532 2.56952.6944
2.82782.96963.1197
2.$x_n$Euler $h=0.1$Euler $h=0.05$Imp. Euler$h=0.1$Imp.
Euler$h=0.05$RK4$h=0.1$RK4$h=0.05$
0.000.050.100.150.200.250.300.350.400.450.500.0000
0.1000
0.2010
0.3049
0.4135
0.5279
0.00000.05000.10010.15060.20170.25370.30670.36100.41670.47390.53270.0000
0.1005
0.2030
0.3092
0.4207
0.53820.00000.05010.10040.15120.20270.25520.30880.36380.42020.47820.53780.0000
0.1003
0.2026
0.3087
0.4201
0.53760.00000.05000.10030.15110.20260.25510.30870.36370.42010.47810.5376
3.$x_n$Euler $h=0.1$Euler $h=0.05$Imp. Euler$h=0.1$Imp.
Euler$h=0.05$RK4$h=0.1$RK4$h=0.05$
0.500.55 0.60 0.65 0.70 0.75 0.800.85 0.900.95 1.00 0.5000
0.6000
0.7095
0.8283
0.9559
1.09210.5000 0.55000.6024 0.65730.7144 0.77390.8356 0.89960.9657
1.03401.10440.5000
0.6048
0.7191
0.8427
0.9752
1.11630.5000 0.55120.6049 0.66090.7193 0.78000.8430 0.90820.9755
1.04511.11680.5000
0.6049
0.7194
0.8431
0.9757
1.11690.5000 0.55120.6049 0.66100.7194 0.78010.8431
0.90830.97571.04521.1169
4. $x_n$Euler $h=0.1$Euler $h=0.05$Imp. Euler$h=0.1$Imp.
Euler$h=0.05$RK4$h=0.1$RK4$h=0.05$
1.001.051.101.151.201.251.301.351.401.451.501.0000
1.2000
1.4760
1.8710
2.4643
3.41651.00001.10001.21831.35951.53001.73891.99882.32842.75673.32964.12531.0000
1.2380
1.5910
2.1524
3.1458
5.25101.00001.10911.24051.40101.60011.85232.17992.61973.23604.15285.64041.0000
1.2415
1.6036
1.6036
3,2745
5.83381.00001.10951.24151.24151.60361.85862,19112.64013,27554.23635.8446
5. Using $y_n+1 = y_n+hu_n$, $y_0 = 3$ $u{n+1} = u_n + h(2x_n+
1)y_n$, $u_0= 1$ we obtain (when $h = 0.2$) $y_1 = y(0.2) = y_0 +
hu_0 = 3 + (0.2)1 = 3.2$. When $h = 0.1$ we have $y_1 = y_0 +
0.1u_0 = 3 + (0.1)1 = 3.1$, $u_1 = u_0 + 0.1 (2x_0 + 1)y_0 = 1 +
0.1(1)3 = 1.3$, $y_2 = y_1 +0.1u_1 = 3.1 + 0.1(1.3) = 3.23$.6. The
first predictor is $y*_3= 1.14822731$. $x_n$ $y_n$
0.0 2.00000000 0.1 1.65620000 0.2 1.41097281 0.3 1.24645047 0.4
1.14796764init. cond.RK4RK4RK4ABM
7. Using $x_0 = 1$, $y_0 = 2$, and $h = 0.1$ we have $x_1 = x_0
+ h(x_0 + y_0) = 1 + 0.1(1 + 2) = 1.3$, $y_1 = y_0 + h(x_0 y_0) = 2
+ 0.1(1 - 2) = 1.9$ and $x_2 = x_1 + h(x_1 + y_1) = 1.3 + 0.1(1.3 +
1.9) = 1.62$, $y_2 = y_1 + h(x_1 y_1) = 1.9 + 0.1(1.3 - 1.9) =
1.84$. Thus, $x(0.2) \approx 1.62$ and $y(0 .2) \approx
1.84$.8.
