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" |
\(3.0\) |
\(1.291345518\) |
\(1.326535737\) |
\(1.344004102\) |
\(1.361383810\) |