A.3.1: Section 3.1 Answers

Last updated
Save as PDF

Page ID: 43757

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

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

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. \(y_{1} = 2.513274123,\: y_{2} = 1.814517822,\: y_{3} = 1.216364496\)

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	Exact
\(1.0\)	\(48.298147362\)	\(51.492825643\)	\(53.076673685\)	\(54.647937102\)

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	Exact
\(2.0\)	\(1.390242009\)	\(1.370996758\)	\(1.361921132\)	\(1.353193719\)

\(x\)	\(h=0.05\)	\(h=0.025\)	\(h=0.0125\)	Exact
\(1.50\)	\(7.886170437\)	\(8.852463793\)	\(9.548039907\)	\(10.500000000\)

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)
\(3.0\)	\(1.469458241\)	\(1.462514486\)	\(1.459217010\)	\(0.3210\)	\(0.1537\)	\(0.0753\)
	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"
\(3.0\)	\(1.291345518\)	\(1.326535737\)	\(1.344004102\)	\(1.361383810\)