1550078481-Ordinary_Differential_Equations__Roberts_

92 Ordinary Differential Equations

Table 2.5 Approximation to the solution of the IVP

(7) y' = y + x; y(O) = 1 on [O, l] with stepsize h = .1

Taylor Improved Fourth

series Euler's Euler 's order Exact

Xn order 3 method method Runge-Kutta solution

.o 1.0 1.0 1.0 1.0 1.0

.1 1.11034 1.1 1.11 1.11034 1.11034

.2 1.24279 1.22 1.2 4205 1.24280 1.24281

.3 1.39969 1.362 1.39846 1.39971 1.39972

.4 1.58361 1.5282 1.58180 1.58364 1.58365

.5 1.79739 1.72102 l. 79489 1.79744 1.79744

.6 2.04417 1.94312 2.04085 2.0 4423 2.04^424

.7 2 .3 2742 2.19743 2.3 2314 2.32750 2.32 751

.8 2.65098 2.48718 2.6455 7 2.65107 2.65 108

.9 3.01908 2.81590 3.01236 3. 01919 3.01921

1.0 3.^43641 3.18748 3. 48215 3.43655 3.43656

Table 2.6 Approximations of the IVP (7) y' = y + x; y(O) = ,1

Euler Improved Euler Runge-Kutta Exact

Xn h = .025 h =. 05 h = .1 solution

.o 1.0 1.0 1.0 1.0

.l 1.10762 1.11025 1.1103 4 1.11034

.2 1.23680 1.24261 1.2 4280 1.24281

.3 1.38977 1.39939 1.39971 1.39972

.4 1.56900 1.5 (^8317) 1. 58364 1.58365
.5 1.77722 1.7967 8 1.79744 1.79744
.6 2.01743 2.04335 2.044 23 2.04424
.7 2 .29297 2.3 2637 2.3 2750 2.32751
.8 2.60749 2.64964 2.65107 2.65108
.9 2.96504 3.017 42 3.01919 3.019 21
1.0 3.37009 3.43437 3.43655 3.43656