A.3.2: Section 3.2 Answers

Last updated
Save as PDF

Page ID: 43758

\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

1. \(y_{1} = 1.542812500,\: y_{2} = 2.421622101,\: y_{3} = 4.208020541\)

2. \(y_{1} = 1.220207973,\: y_{2} = 1.489578775.\: y_{3} = 1.819337186\)

3. \(y_{1} = 1.890687500,\: y_{2} = 1.763784003,\: y_{3} = 1.622698378\)

4. \(y_{1} = 2.961317914,\: y_{2} = 2.920132727,\: y_{3} = 2.876213748\)

5. \(y_{1} = 2.478055238,\: y_{2} = 1.844042564,\: y_{3} = 1.313882333\)

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	Exact
\(1.0\)	\(56.134480009\)	\(55.003390448\)	\(54.734674836\)	\(54.647937102\)

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	Exact
\(2.0\)	\(1.353501839\)	\(1.353288493\)	\(1.353219485\)	\(1.353193719\)

\(x\)	\(h=0.5\)	\(h=0.025\)	\(h=0.0125\)	Exact
\(1.50\)	\(10.141969585\)	\(10.396770409\)	\(10.472502111\)	\(10.500000000\)

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)
\(3.0\)	\(1.455674816\)	\(1.455935127\)	\(1.456001289\)	\(-0.00818\)	\(-0.00207\)	\(-0.000518\)
	Approximate Solutions			Residuals

10.

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)
\(2.0\)	\(0.492862999\)	\(0.492709931\)	\(0.492674855\)	\(0.00335\)	\(0.000777\)	\(0.000187\)
	Approximate Solutions			Residuals

11.

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	"Exact"
\(1.0\)	\(0.660268159\)	\(0.660028505\)	\(0.659974464\)	\(0.659957689\)

12.

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	"Exact"
\(2.0\)	\(-0.749751364\)	\(-0.750637632\)	\(-0.750845571\)	\(-0.750912371\)

13. Applying variation of parameters to the given initial value problem \(y = ue^{−3x}\), where \((A) u' = 1 − 2x, u(0) = 2\). Since \(u''' = 0\), the improved Euler method yields the exact solution of (A). Therefore the improved Euler semilinear method produces the exact solution of the given problem.

Improved Euler method
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	Exact
\(1.0\)	\(0.105660401\)	\(0.100924399\)	\(0.099893685\)	\(0.099574137\)

Improved Euler semilinar method
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	Exact
\(1.0\)	\(0.099574137\)	\(0.099574137\)	\(0.099574137\)	\(0.099574137\)

14.

Improved Euler method
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	"Exact"
\(3.0\)	\(15.107600968\)	\(15.234856000\)	\(15.269755072\)	\(15.282004826\)

Improved Euler semilinar method
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	"Exact"
\(3.0\)	\(15.285231726\)	\(15.282812424\)	\(15.282206780\)	\(15.282004826\)

15.

Improved Euler method
\(x\)	\(h=0.2\)	\(h=0.1\)	\(h=0.05\)	"Exact"
\(2.0\)	\(0.924335375\)	\(0.907866081\)	\(0.905058201\)	\(0.904276722\)

Improved Euler semilinear method
\(x\)	\(h=0.2\)	\(h=0.1\)	\(h=0.05\)	"Exact"
\(2.0\)	\(0.969670789\)	\(0.920861858\)	\(0.908438261\)	\(0.904276722\)

16.

Improved Euler method
\(x\)	\(h=0.2\)	\(h=0.1\)	\(h=0.05\)	"Exact"
\(3.0\)	\(0.967473721\)	\(0.967510790\)	\(0.967520062\)	\(0.967523153\)

Improved Euler semilinear method
\(x\)	\(h=0.2\)	\(h=0.1\)	\(h=0.05\)	"Exact"
\(3.0\)	\(0.967473721\)	\(0.967510790\)	\(0.967520062\)	\(0.967523153\)

17.

Improved Euler method
\(x\)	\(h=0.0500\)	\(h=0.0250\)	\(h=0.0125\)	"Exact"
\(1.50\)	\(0.349176060\)	\(0.345171664\)	\(0.344131282\)	\(0.343780513\)

Improved Euler semilinear method
\(x\)	\(h=0.0500\)	\(h=0.0250\)	\(h=0.0125\)	"Exact"
\(1.50\)	\(0.349350206\)	\(0.345216894\)	\(0.344142832\)	\(0.343780513\)

18.

Improved Euler method
\(x\)	\(h=0.2\)	\(h=0.1\)	\(h=0.05\)	"Exact"
\(2.0\)	\(0.732679223\)	\(0.732721613\)	\(0.732667905\)	\(0.732638628\)

Improved Euler semilinear method
\(x\)	\(h=0.2\)	\(h=0.1\)	\(h=0.05\)	"Exact"
\(2.0\)	\(0.732166678\)	\(0.732521078\)	\(0.732609267\)	\(0.732638628\)

19.

Improved Euler method
\(x\)	\(h=0.0500\)	\(h=0.0250\)	\(h=0.0125\)	"Exact"
\(1.50\)	\(2.247880315\)	\(2.244975181\)	\(2.244260143\)	\(2.244023982\)

Improved Euler semilinear method
\(x\)	\(h=0.0500\)	\(h=0.0250\)	\(h=0.0125\)	"Exact"
\(1.50\)	\(2.248603585\)	\(2.245169707\)	\(2.244310465\)	\(2.244023982\)

20.

Improved Euler method
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	"Exact"
\(1.0\)	\(0.059071894\)	\(0.056999028\)	\(0.056553023\)	\(0.056415515\)

Improved Euler semilinear method
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	"Exact"
\(1.0\)	\(0.056295914\)	\(0.056385765\)	\(0.056408124\)	\(0.056415515\)

21.

Improved Euler method
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	"Exact"
\(1.0\)	\(50.534556346\)	\(53.483947013\)	\(54.391544440\)	\(54.729594761\)

Improved Euler semilinear method
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	"Exact"
\(1.0\)	\(54.709041434\)	\(54.724083572\)	\(54.728191366\)	\(54.729594761\)

22.

Improved Euler method
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	"Exact"
\(3.0\)	\(1.361395309\)	\(1.361379259\)	\(1.361382239\)	\(1.361383810\)

Improved Euler semilinear method
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	"Exact"
\(3.0\)	\(1.375699933\)	\(1.364730937\)	\(1.362193997\)	\(1.361383810\)

23.

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	Exact
\(2.0\)	\(1.349489056\)	\(1.352345900\)	\(1.352990822\)	\(1.353193719\)

24.

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	Exact
\(2.0\)	\(1.350890736\)	\(1.352667599\)	\(1.353067951\)	\(1.353193719\)

25.

\(x\)	\(h=0.05\)	\(h=0.025\)	\(h=0.0125\)	Exact
\(1.50\)	\(10.133021311\)	\(10.391655098\)	\(10.470731411\)	\(10.500000000\)

26.

\(x\)	\(h=0.05\)	\(h=0.025\)	\(h=0.0125\)	Exact
\(1.50\)	\(10.136329642\)	\(10.393419681\)	\(10.470731411\)	\(10.500000000\)

27.

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	"Exact"
\(1.0\)	\(0.660846835\)	\(0.660189749\)	\(0.660016904\)	\(0.659957689\)

28.

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	"Exact"
\(1.0\)	\(0.660658411\)	\(0.660136630\)	\(0.660002840\)	\(0.659957689\)

29.

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	"Exact"
\(2.0\)	\(-0.750626284\)	\(-0.750844513\)	\(-0.750895864\)	\(-0.751331499\)

30.

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	"Exact"
\(2.0\)	\(-0.750335016\)	\(-0.750775571\)	\(-0.750879100\)	\(-0.751331499\)