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A.3.2: Section 3.2 Answers

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Apr 25, 2022
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- A.3.1: Section 3.1 Answers
- A.3.3: Section 3.3 Answers

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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"
$2.0$	$-0.750335016$	$-0.750775571$	$-0.750879100$	$-0.751331499$

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