14.27: Sección 3.2 Respuestas

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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. \(y_{1} = 2.478055238,\: y_{2} = 1.844042564,\: y_{3} = 1.313882333\)

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	Exacto
\(1.0\)	\(56.134480009\)	\(55.003390448\)	\(54.734674836\)	\(54.647937102\)

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	Exacto
\(2.0\)	\(1.353501839\)	\(1.353288493\)	\(1.353219485\)	\(1.353193719\)

\(x\)	\(h=0.5\)	\(h=0.025\)	\(h=0.0125\)	Exacto
\(1.50\)	\(10.141969585\)	\(10.396770409\)	\(10.472502111\)	\(10.500000000\)

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)
\(3.0\)	\(1.455674816\)	\(1.455935127\)	\(1.456001289\)	\(-0.00818\)	\(-0.00207\)	\(-0.000518\)
	Soluciones aproximadas			Residuales

10.

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)
\(2.0\)	\(0.492862999\)	\(0.492709931\)	\(0.492674855\)	\(0.00335\)	\(0.000777\)	\(0.000187\)
	Soluciones aproximadas			Residuales

11.

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	“Exacto”
\(1.0\)	\(0.660268159\)	\(0.660028505\)	\(0.659974464\)	\(0.659957689\)

12.

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	“Exacto”
\(2.0\)	\(-0.749751364\)	\(-0.750637632\)	\(-0.750845571\)	\(-0.750912371\)

13. Aplicando variación de parámetros al problema de valor inicial dado\(y = ue^{−3x}\), donde\((A) u' = 1 − 2x, u(0) = 2\). Ya que\(u''' = 0\), el método mejorado de Euler produce la solución exacta de (A). Por lo tanto, el método semilineal mejorado de Euler produce la solución exacta del problema dado.

Método Euler mejorado
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	Exacto
\(1.0\)	\(0.105660401\)	\(0.100924399\)	\(0.099893685\)	\(0.099574137\)

Método semilinar de Euler mejorado
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	Exacto
\(1.0\)	\(0.099574137\)	\(0.099574137\)	\(0.099574137\)	\(0.099574137\)

14.

Método Euler mejorado
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	“Exacto”
\(3.0\)	\(15.107600968\)	\(15.234856000\)	\(15.269755072\)	\(15.282004826\)

Método semilinar de Euler mejorado
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	“Exacto”
\(3.0\)	\(15.285231726\)	\(15.282812424\)	\(15.282206780\)	\(15.282004826\)

15.

Método Euler mejorado
\(x\)	\(h=0.2\)	\(h=0.1\)	\(h=0.05\)	“Exacto”
\(2.0\)	\(0.924335375\)	\(0.907866081\)	\(0.905058201\)	\(0.904276722\)

Método semilineal de Euler mejorado
\(x\)	\(h=0.2\)	\(h=0.1\)	\(h=0.05\)	“Exacto”
\(2.0\)	\(0.969670789\)	\(0.920861858\)	\(0.908438261\)	\(0.904276722\)

16.

Método Euler mejorado
\(x\)	\(h=0.2\)	\(h=0.1\)	\(h=0.05\)	“Exacto”
\(3.0\)	\(0.967473721\)	\(0.967510790\)	\(0.967520062\)	\(0.967523153\)

Método semilineal de Euler mejorado
\(x\)	\(h=0.2\)	\(h=0.1\)	\(h=0.05\)	“Exacto”
\(3.0\)	\(0.967473721\)	\(0.967510790\)	\(0.967520062\)	\(0.967523153\)

17.

Método Euler mejorado
\(x\)	\(h=0.0500\)	\(h=0.0250\)	\(h=0.0125\)	“Exacto”
\(1.50\)	\(0.349176060\)	\(0.345171664\)	\(0.344131282\)	\(0.343780513\)

Método semilineal de Euler mejorado
\(x\)	\(h=0.0500\)	\(h=0.0250\)	\(h=0.0125\)	“Exacto”
\(1.50\)	\(0.349350206\)	\(0.345216894\)	\(0.344142832\)	\(0.343780513\)

18.

Método Euler mejorado
\(x\)	\(h=0.2\)	\(h=0.1\)	\(h=0.05\)	“Exacto”
\(2.0\)	\(0.732679223\)	\(0.732721613\)	\(0.732667905\)	\(0.732638628\)

Método semilineal de Euler mejorado
\(x\)	\(h=0.2\)	\(h=0.1\)	\(h=0.05\)	“Exacto”
\(2.0\)	\(0.732166678\)	\(0.732521078\)	\(0.732609267\)	\(0.732638628\)

19.

Método Euler mejorado
\(x\)	\(h=0.0500\)	\(h=0.0250\)	\(h=0.0125\)	“Exacto”
\(1.50\)	\(2.247880315\)	\(2.244975181\)	\(2.244260143\)	\(2.244023982\)

Método semilineal de Euler mejorado
\(x\)	\(h=0.0500\)	\(h=0.0250\)	\(h=0.0125\)	“Exacto”
\(1.50\)	\(2.248603585\)	\(2.245169707\)	\(2.244310465\)	\(2.244023982\)

20.

Método Euler mejorado
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	“Exacto”
\(1.0\)	\(0.059071894\)	\(0.056999028\)	\(0.056553023\)	\(0.056415515\)

Método semilineal de Euler mejorado
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	“Exacto”
\(1.0\)	\(0.056295914\)	\(0.056385765\)	\(0.056408124\)	\(0.056415515\)

21.

Método Euler mejorado
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	“Exacto”
\(1.0\)	\(50.534556346\)	\(53.483947013\)	\(54.391544440\)	\(54.729594761\)

Método semilineal de Euler mejorado
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	“Exacto”
\(1.0\)	\(54.709041434\)	\(54.724083572\)	\(54.728191366\)	\(54.729594761\)

22.

Método Euler mejorado
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	“Exacto”
\(3.0\)	\(1.361395309\)	\(1.361379259\)	\(1.361382239\)	\(1.361383810\)

Método semilineal de Euler mejorado
\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	“Exacto”
\(3.0\)	\(1.375699933\)	\(1.364730937\)	\(1.362193997\)	\(1.361383810\)

23.

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	Exacto
\(2.0\)	\(1.349489056\)	\(1.352345900\)	\(1.352990822\)	\(1.353193719\)

24.

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	Exacto
\(2.0\)	\(1.350890736\)	\(1.352667599\)	\(1.353067951\)	\(1.353193719\)

25.

\(x\)	\(h=0.05\)	\(h=0.025\)	\(h=0.0125\)	Exacto
\(1.50\)	\(10.133021311\)	\(10.391655098\)	\(10.470731411\)	\(10.500000000\)

26.

\(x\)	\(h=0.05\)	\(h=0.025\)	\(h=0.0125\)	Exacto
\(1.50\)	\(10.136329642\)	\(10.393419681\)	\(10.470731411\)	\(10.500000000\)

27.

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	“Exacto”
\(1.0\)	\(0.660846835\)	\(0.660189749\)	\(0.660016904\)	\(0.659957689\)

28.

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	“Exacto”
\(1.0\)	\(0.660658411\)	\(0.660136630\)	\(0.660002840\)	\(0.659957689\)

29.

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	“Exacto”
\(2.0\)	\(-0.750626284\)	\(-0.750844513\)	\(-0.750895864\)	\(-0.751331499\)

30.

\(x\)	\(h=0.1\)	\(h=0.05\)	\(h=0.025\)	“Exacto”
\(2.0\)	\(-0.750335016\)	\(-0.750775571\)	\(-0.750879100\)	\(-0.751331499\)