A.3.2: Section 3.2 Answers
- Page ID
- 43758
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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. \(y_{1} = 1.542812500,\: y_{2} = 2.421622101,\: y_{3} = 4.208020541\)
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. \(y_{1} = 2.478055238,\: y_{2} = 1.844042564,\: y_{3} = 1.313882333\)
6.
| \(x\) | \(h=0.1\) | \(h=0.05\) | \(h=0.025\) | Exact |
| \(1.0\) | \(56.134480009\) | \(55.003390448\) | \(54.734674836\) | \(54.647937102\) |
7.
| \(x\) | \(h=0.1\) | \(h=0.05\) | \(h=0.025\) | Exact |
| \(2.0\) | \(1.353501839\) | \(1.353288493\) | \(1.353219485\) | \(1.353193719\) |
8.
| \(x\) | \(h=0.5\) | \(h=0.025\) | \(h=0.0125\) | Exact |
| \(1.50\) | \(10.141969585\) | \(10.396770409\) | \(10.472502111\) | \(10.500000000\) |
9.
| \(x\) | \(h=0.1\) | \(h=0.05\) | \(h=0.025\) | \(h=0.1\) | \(h=0.05\) | \(h=0.025\) |
| \(3.0\) | \(1.455674816\) | \(1.455935127\) | \(1.456001289\) | \(-0.00818\) | \(-0.00207\) | \(-0.000518\) |
| 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" |
| \(2.0\) | \(-0.750335016\) | \(-0.750775571\) | \(-0.750879100\) | \(-0.751331499\) |