A.3.1: Section 3.1 Answers

1. $y_{1}= 1.450000000,\: y_{2} = 2.085625000,\: y_{3} = 3.079099746$

2. $y_{1} = 1.200000000,\: y_{2} = 1.440415946,\: y_{3} = 1.729880994$

3. $y_{1} = 1.900000000,\: y_{2} = 1.781375000,\: y_{3} = 1.646612970$

4. $y_{1} = 2.962500000,\: y_{2} = 2.922635828,\: y_{3} = 2.880205639$

5. $y_{1} = 2.513274123,\: y_{2} = 1.814517822,\: y_{3} = 1.216364496$

6.

$x$	$h=0.1$	$h=0.05$	$h=0.025$	Exact
$1.0$	$48.298147362$	$51.492825643$	$53.076673685$	$54.647937102$

7.

$x$	$h=0.1$	$h=0.05$	$h=0.025$	Exact
$2.0$	$1.390242009$	$1.370996758$	$1.361921132$	$1.353193719$

8.

$x$	$h=0.05$	$h=0.025$	$h=0.0125$	Exact
$1.50$	$7.886170437$	$8.852463793$	$9.548039907$	$10.500000000$

9.

$x$	$h=0.1$	$h=0.05$	$h=0.025$	$h=0.1$	$h=0.05$	$h=0.025$
$3.0$	$1.469458241$	$1.462514486$	$1.459217010$	$0.3210$	$0.1537$	$0.0753$
	Approximate Solutions			Residuals

10.

$x$	$h=0.1$	$h=0.05$	$h=0.025$	$h=0.1$	$h=0.05$	$h=0.025$
$2.0$	$0.473456737$	$0.483227470$	$0.487986391$	$-0.3129$	$-0.1563$	$-0.0781$
	Approximate Solutions			Residuals

11.

$x$	$h=0.1$	$h=0.05$	$h=0.025$	"Exact"
$1.0$	$0.691066797$	$0.676269516$	$0.668327471$	$0.659957689$

12.

$x$	$h=0.1$	$h=0.05$	$h=0.025$	"Exact"
$2.0$	$-0.772381768$	$-0.761510960$	$-0.756179726$	$-0.750912371$

13.

Euler's Method
$x$	$h=0.1$	$h=0.05$	$h=0.025$	Exact
$1.0$	$0.538871178$	$0.593002325$	$0.620131525$	$0.647231889$

Euler semilinear Method
$x$	$h=0.1$	$h=0.05$	$h=0.025$	Exact
$1.0$	$0.647231889$	$0.647231889$	$0.647231889$	$0.647231889$

Applying variation of parameters to the given initial value problem yields $y = ue^{−3x}$, where (A) $u' = 7, u(0) = 6$. Since $u''= 0$, Euler’s method yields the exact solution of (A). Therefore the Euler semilinear method produces the exact solution of the given problem

14.

Euler's Method
$x$	$h=0.1$	$h=0.05$	$h=0.025$	"Exact"
$3.0$	$12.804226135$	$13.912944662$	$14.559623055$	$15.282004826$

Euler semilinear method
$x$	$h=0.1$	$h=0.05$	$h=0.025$	"Exact"
$3.0$	$15.354122287$	$15.317257705$	$15.299429421$	$15.282004826$

15.

Euler's method
$x$	$h=0.2$	$h=0.1$	$h=0.05$	"Exact"
$2.0$	$0.867565004$	$0.885719263$	$0.895024772$	$0.904276722$

Euler's semilinear method
$x$	$h=0.2$	$h=0.1$	$h=0.05$	"Exact"
$2.0$	$0.569670789$	$0.720861858$	$0.808438261$	$0.904276722$

16.

Euler's method
$x$	$h=0.2$	$h=0.1$	$h=0.05$	"Exact"
$3.0$	$0.922094379$	$0.945604800$	$0.956752868$	$0.967523153$

Euler semilinear method
$x$	$h=0.2$	$h=0.1$	$h=0.05$	"Exact"
$3.0$	$0.993954754$	$0.980751307$	$0.974140320$	$0.967523153$

17.

Euler's method
$x$	$h=0.0500$	$h=0.0250$	$h=0.0125$	"Exact"
$1.50$	$0.319892131$	$0.330797109$	$0.337020123$	$0.343780513$

Euler semilinear method
$x$	$h=0.0500$	$h=0.0250$	$h=0.0125$	"Exact"
$1.50$	$0.305596953$	$0.323340268$	$0.333204519$	$0.343780513$

18.

Euler's method
$x$	$h=0.2$	$h=0.1$	$h=0.05$	"Exact"
$2.0$	$0.754572560$	$0.743869878$	$0.738303914$	$0.732638628$

Euler semilinear method
$x$	$h=0.2$	$h=0.1$	$h=0.05$	"Exact"
$2.0$	$0.722610454$	$0.727742966$	$0.730220211$	$0.732638628$

19.

Euler's method
$x$	$h=0.0500$	$h=0.0250$	$h=0.0125$	"Exact"
$1.50$	$2.175959970$	$2.210259554$	$2.227207500$	$2.244023982$

Euler semilinear method
$x$	$h=0.0500$	$h=0.0250$	$h=0.0125$	"Exact"
$1.50$	$2.117953342$	$2.179844585$	$2.211647904$	$2.244023982$

20.

Euler's method
$x$	$h=0.1$	$h=0.05$	$h=0.025$	"Exact"
$1.0$	$0.032105117$	$0.043997045$	$0.050159310$	$0.056415515$

Euler's semilinear method
$x$	$h=0.1$	$h=0.05$	$h=0.025$	"Exact"
$1.0$	$0.056020154$	$0.056243980$	$0.056336491$	$0.056415515$

21.

Euler's method
$x$	$h=0.1$	$h=0.05$	$h=0.025$	"Exact"
$1.0$	$28.987816656$	$38.426957516$	$45.367269688$	$54.729594761$

Euler's semilinar method
$x$	$h=0.1$	$h=0.05$	$h=0.025$	"Exact"
$1.0$	$54.709134946$	$54.724150485$	$54.728228015$	$54.729594761$

22.

Euler's method
$x$	$h=0.1$	$h=0.05$	$h=0.025$	"Exact"
$3.0$	$1.361427907$	$1.361320824$	$1.361332589$	$1.361383810$

Euler's semilinar method
$x$	$h=0.1$	$h=0.05$	$h=0.025$	"Exact"
$3.0$	$1.291345518$	$1.326535737$	$1.344004102$	$1.361383810$