A.3.1: Section 3.1 Answers
- Page ID
- 43757
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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" |
| \(3.0\) | \(1.291345518\) | \(1.326535737\) | \(1.344004102\) | \(1.361383810\) |