36
Valores Iniciales: Tiempo incial: 0 Tiempo final: 20 Posición inici 0 Velocidad incia0 Tiempo T1 T2 T3 T4 X1 0.05 0.05 0.06 0.06 0.07 0 0.1 0.1 0.11 0.11 0.12 0.005 0.15 0.15 0.16 0.16 0.17 0.01962656 0.2 0.2 0.21 0.21 0.22 0.04315145 0.25 0.25 0.26 0.26 0.27 0.07440743 0.3 0.3 0.31 0.31 0.32 0.11184624 0.35 0.35 0.36 0.36 0.37 0.15361522 0.4 0.4 0.41 0.41 0.42 0.19764904 0.45 0.45 0.46 0.46 0.47 0.24177177 0.5 0.5 0.51 0.51 0.52 0.28380447 0.55 0.55 0.56 0.56 0.57 0.32167272 0.6 0.6 0.61 0.61 0.62 0.35350906 0.65 0.65 0.66 0.66 0.67 0.37774504 0.7 0.7 0.71 0.71 0.72 0.39318852 0.75 0.75 0.76 0.76 0.77 0.39908227 0.8 0.8 0.81 0.81 0.82 0.39514114 0.85 0.85 0.86 0.86 0.87 0.38156582 0.9 0.9 0.91 0.91 0.92 0.35903265 0.95 0.95 0.96 0.96 0.97 0.32865991 1 1 1.01 1.01 1.02 0.29195233 1.05 1.05 1.06 1.06 1.07 0.25072653 1.1 1.1 1.11 1.11 1.12 0.20702117 1.15 1.15 1.16 1.16 1.17 0.16299605 1.2 1.2 1.21 1.21 1.22 0.12082546 1.25 1.25 1.26 1.26 1.27 0.08259076 1.3 1.3 1.31 1.31 1.32 0.05017766 1.35 1.35 1.36 1.36 1.37 0.02518321 1.4 1.4 1.41 1.41 1.42 0.00883714 1.45 1.45 1.46 1.46 1.47 0.00194131 1.5 1.5 1.51 1.51 1.52 0.00483043 1.55 1.55 1.56 1.56 1.57 0.01735579 1.6 1.6 1.61 1.61 1.62 0.0388929 1.65 1.65 1.66 1.66 1.67 0.06837265 1.7 1.7 1.71 1.71 1.72 0.10433433 1.75 1.75 1.76 1.76 1.77 0.14499805 1.8 1.8 1.81 1.81 1.82 0.18835283 1.85 1.85 1.86 1.86 1.87 0.23225602 1.9 1.9 1.91 1.91 1.92 0.27453922 1.95 1.95 1.96 1.96 1.97 0.31311537 2 2 2.01 2.01 2.02 0.34608175 2.05 2.05 2.06 2.06 2.07 0.37181386 2.1 2.1 2.11 2.11 2.12 0.38904544 2.15 2.15 2.16 2.16 2.17 0.39693078 2.2 2.2 2.21 2.21 2.22 0.39508626 2.25 2.25 2.26 2.26 2.27 0.38360898 2.3 2.3 2.31 2.31 2.32 0.36307165 2.35 2.35 2.36 2.36 2.37 0.33449405 0 1 -1 0 1 PO

Ruge Kutta

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Hoja de excel para calcular Runge Kutta

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Page 1: Ruge Kutta

Valores Iniciales: Tiempo incial: 0 Tiempo final: 20 h=Posición inicial: 0 Velocidad incial: 0

Tiempo T1 T2 T3 T4 X1 X20.05 0.05 0.06 0.06 0.07 0 00.1 0.1 0.11 0.11 0.12 0.005 0.0099375

0.15 0.15 0.16 0.16 0.17 0.01962656 0.029254690.2 0.2 0.21 0.21 0.22 0.04315145 0.05699184

0.25 0.25 0.26 0.26 0.27 0.07440743 0.091774010.3 0.3 0.31 0.31 0.32 0.11184624 0.13187922

0.35 0.35 0.36 0.36 0.37 0.15361522 0.175323680.4 0.4 0.41 0.41 0.42 0.19764904 0.21995991

0.45 0.45 0.46 0.46 0.47 0.24177177 0.263582910.5 0.5 0.51 0.51 0.52 0.28380447 0.30403908

0.55 0.55 0.56 0.56 0.57 0.32167272 0.339332550.6 0.6 0.61 0.61 0.62 0.35350906 0.3677236

0.65 0.65 0.66 0.66 0.67 0.37774504 0.387814460.7 0.7 0.71 0.71 0.72 0.39318852 0.39861812

0.75 0.75 0.76 0.76 0.77 0.39908227 0.399606790.8 0.8 0.81 0.81 0.82 0.39514114 0.3907377

0.85 0.85 0.86 0.86 0.87 0.38156582 0.372454920.9 0.9 0.91 0.91 0.92 0.35903265 0.34566712

0.95 0.95 0.96 0.96 0.97 0.32865991 0.31170241 1 1.01 1.01 1.02 0.29195233 0.27224246

1.05 1.05 1.06 1.06 1.07 0.25072653 0.229239341.1 1.1 1.11 1.11 1.12 0.20702117 0.18481884

1.15 1.15 1.16 1.16 1.17 0.16299605 0.141175451.2 1.2 1.21 1.21 1.22 0.12082546 0.10046391

1.25 1.25 1.26 1.26 1.27 0.08259076 0.064692871.3 1.3 1.31 1.31 1.32 0.05017766 0.03562576

1.35 1.35 1.36 1.36 1.37 0.02518321 0.014693851.4 1.4 1.41 1.41 1.42 0.00883714 0.00292579

1.45 1.45 1.46 1.46 1.47 0.00194131 0.000897081.5 1.5 1.51 1.51 1.52 0.00483043 0.00870188

1.55 1.55 1.56 1.56 1.57 0.01735579 0.025948711.6 1.6 1.61 1.61 1.62 0.0388929 0.05178003

1.65 1.65 1.66 1.66 1.67 0.06837265 0.084914931.7 1.7 1.71 1.71 1.72 0.10433433 0.1237126

1.75 1.75 1.76 1.76 1.77 0.14499805 0.166253611.8 1.8 1.81 1.81 1.82 0.18835283 0.21043486

1.85 1.85 1.86 1.86 1.87 0.23225602 0.254073541.9 1.9 1.91 1.91 1.92 0.27453922 0.29501498

1.95 1.95 1.96 1.96 1.97 0.31311537 0.331239052 2 2.01 2.01 2.02 0.34608175 0.3609598

2.05 2.05 2.06 2.06 2.07 0.37181386 0.382713572.1 2.1 2.11 2.11 2.12 0.38904544 0.395431

2.15 2.15 2.16 2.16 2.17 0.39693078 0.398489642.2 2.2 2.21 2.21 2.22 0.39508626 0.39174443

2.25 2.25 2.26 2.26 2.27 0.38360898 0.375534492.3 2.3 2.31 2.31 2.32 0.36307165 0.35066618

2.35 2.35 2.36 2.36 2.37 0.33449405 0.318372892.4 2.4 2.41 2.41 2.42 0.29929241 0.28025395

0 1 2 3 4 5

-1

0

1

POSICION Y VELOCIDAD VS TIEMPO

POSICIÓN VELOCIDAD

Page 2: Ruge Kutta

2.45 2.45 2.46 2.46 2.47 0.25920914 0.238195372.5 2.5 2.51 2.51 2.52 0.21622671 0.19427655

2.55 2.55 2.56 2.56 2.57 0.17246948 0.150667482.6 2.6 2.61 2.61 2.62 0.13009879 0.1095215

2.65 2.65 2.66 2.66 2.67 0.09120617 0.0728692.7 2.7 2.71 2.71 2.72 0.0577101 0.04251721

2.75 2.75 2.76 2.76 2.77 0.03126138 0.019961092.8 2.8 2.81 2.81 2.82 0.01316177 0.00630972

2.85 2.85 2.86 2.86 2.87 0.00429993 0.002231752.9 2.9 2.91 2.91 2.92 0.00510778 0.00792265

2.95 2.95 2.96 2.96 2.97 0.01553945 0.023095353 3 3.01 3.01 3.02 0.03507387 0.04699474

3.05 3.05 3.06 3.06 3.07 0.06274075 0.078435213.1 3.1 3.11 3.11 3.12 0.0971688 0.11585949

3.15 3.15 3.16 3.16 3.17 0.13665369 0.157415763.2 3.2 3.21 3.21 3.22 0.17924244 0.20104932

3.25 3.25 3.26 3.26 3.27 0.22282996 0.244604123.3 3.3 3.31 3.31 3.32 0.26526318 0.28592937

3.35 3.35 3.36 3.36 3.37 0.30444733 0.322985683.4 3.4 3.41 3.41 3.42 0.33844939 0.35394574

3.45 3.45 3.46 3.46 3.47 0.36559344 0.377284413.5 3.5 3.51 3.51 3.52 0.38454327 0.39185388

3.55 3.55 3.56 3.56 3.57 0.39436815 0.396940083.6 3.6 3.61 3.61 3.62 0.39458851 0.39229768

3.65 3.65 3.66 3.66 3.67 0.3851994 0.378161943.7 3.7 3.71 3.71 3.72 0.36667043 0.35523679

3.75 3.75 3.76 3.76 3.77 0.33992224 0.324659763.8 3.8 3.81 3.81 3.82 0.3062808 0.28794556

3.85 3.85 3.86 3.86 3.87 0.26741168 0.246911013.9 3.9 3.91 3.91 3.92 0.22523757 0.20358519

3.95 3.95 3.96 3.96 3.97 0.18184319 0.160109074 4 4.01 4.01 4.02 0.13937222 0.1186297

4.05 4.05 4.06 4.06 4.07 0.09992144 0.081194234.1 4.1 4.11 4.11 4.12 0.06543717 0.04964883

4.15 4.15 4.16 4.16 4.17 0.03761927 0.025547664.2 4.2 4.21 4.21 4.22 0.01783736 0.01007631

4.25 4.25 4.26 4.26 4.27 0.00706336 0.003993484.3 4.3 4.31 4.31 4.32 0.00582376 0.00759375

4.35 4.35 4.36 4.36 4.37 0.0141739 0.020693374.4 4.4 4.41 4.41 4.42 0.0316955 0.04263956

4.45 4.45 4.46 4.46 4.47 0.05751764 0.072343114.5 4.5 4.51 4.51 4.52 0.09036001 0.10833239

4.55 4.55 4.56 4.56 4.57 0.12859645 0.148826254.6 4.6 4.61 4.61 4.62 0.1703354 0.19182223

4.65 4.65 4.66 4.66 4.67 0.21351351 0.235195524.7 4.7 4.71 4.71 4.72 0.25599763 0.27680397

4.75 4.75 4.76 4.76 4.77 0.29569022 0.314593964.8 4.8 4.81 4.81 4.82 0.33063285 0.34670165

4.85 4.85 4.86 4.86 4.87 0.3591029 0.371544984.9 4.9 4.91 4.91 4.92 0.37969843 0.3879016

4.95 4.95 4.96 4.96 4.97 0.39140723 0.39496902

Page 3: Ruge Kutta

5 5 5.01 5.01 5.02 0.39365658 0.392403965.05 5.05 5.06 5.06 5.07 0.38634124 0.380339045.1 5.1 5.11 5.11 5.12 0.36982836 0.35937591

5.15 5.15 5.16 5.16 5.17 0.34493906 0.330555295.2 5.2 5.21 5.21 5.22 0.31290763 0.29530527

5.25 5.25 5.26 5.26 5.27 0.27532027 0.255370565.3 5.3 5.31 5.31 5.32 0.23403658 0.21272615

5.35 5.35 5.36 5.36 5.37 0.19109755 0.16947965.4 5.4 5.41 5.41 5.42 0.14862465 0.12776693

5.45 5.45 5.46 5.46 5.47 0.10871503 0.089647075.5 5.5 5.51 5.51 5.52 0.07333794 0.05700028

5.55 5.55 5.56 5.56 5.57 0.04423762 0.031435375.6 5.6 5.61 5.61 5.62 0.02284724 0.01421043

5.65 5.65 5.66 5.66 5.67 0.01021836 0.006170935.7 5.7 5.71 5.71 5.72 0.00696923 0.00770822

5.75 5.75 5.76 5.76 5.77 0.01325451 0.018740475.8 5.8 5.81 5.81 5.82 0.02875793 0.03871702

5.85 5.85 5.86 5.86 5.87 0.05270819 0.066645845.9 5.9 5.91 5.91 5.92 0.08391735 0.10114283

5.95 5.95 5.96 5.96 5.97 0.12083974 0.140500386 6 6.01 6.01 6.02 0.1616485 0.18277189

6.05 6.05 6.06 6.06 6.07 0.20432598 0.225868086.1 6.1 6.11 6.11 6.12 0.24676349 0.26766024

6.15 6.15 6.16 6.16 6.17 0.28686549 0.306085356.2 6.2 6.21 6.21 6.22 0.32265308 0.33924795

6.25 6.25 6.26 6.26 6.27 0.35236163 0.365513636.3 6.3 6.31 6.31 6.32 0.3745278 0.38358957

6.35 6.35 6.36 6.36 6.37 0.38806161 0.392588196.4 6.4 6.41 6.41 6.42 0.39230006 0.3920707

6.45 6.45 6.46 6.46 6.47 0.38703958 0.382068546.5 6.5 6.51 6.51 6.52 0.3725458 0.3630815

6.55 6.55 6.56 6.56 6.57 0.34954017 0.336052756.6 6.6 6.61 6.61 6.62 0.31916402 0.30232202

6.65 6.65 6.66 6.66 6.67 0.28292197 0.263559166.7 6.7 6.71 6.71 6.72 0.24260737 0.22168149

6.75 6.75 6.76 6.76 6.77 0.20021357 0.178758976.8 6.8 6.81 6.81 6.82 0.1578354 0.13691189

6.85 6.85 6.86 6.86 6.87 0.11756559 0.09820616.9 6.9 6.91 6.91 6.92 0.08139147 0.06455107

6.95 6.95 6.96 6.96 6.97 0.0510969 0.037605667 7 7.01 7.01 7.02 0.02817429 0.01869637

7.05 7.05 7.06 7.06 7.07 0.01375105 0.008752047.1 7.1 7.11 7.11 7.12 0.00853422 0.0082582

7.15 7.15 7.16 7.16 7.17 0.01277573 0.017233427.2 7.2 7.21 7.21 7.22 0.02626028 0.03522862

7.25 7.25 7.26 7.26 7.27 0.04831626 0.06134967.3 7.3 7.31 7.31 7.32 0.0778492 0.0943014

7.35 7.35 7.36 7.36 7.37 0.11339606 0.132452557.4 7.4 7.41 7.41 7.42 0.15319773 0.17391584

7.45 7.45 7.46 7.46 7.47 0.19528606 0.216641657.5 7.5 7.51 7.51 7.52 0.23758121 0.25851928

Page 4: Ruge Kutta

7.55 7.55 7.56 7.56 7.57 0.27799438 0.297481227.6 7.6 7.61 7.61 7.62 0.31453103 0.33160521

7.65 7.65 7.66 7.66 7.67 0.34538926 0.359209117.7 7.7 7.71 7.71 7.72 0.36904872 0.37893377

7.75 7.75 7.76 7.76 7.77 0.38434548 0.389810017.8 7.8 7.81 7.81 7.82 0.39052929 0.39130618

7.85 7.85 7.86 7.86 7.87 0.38730042 0.38335427.9 7.9 7.91 7.91 7.92 0.37482412 0.36635257

7.95 7.95 7.96 7.96 7.97 0.35372221 0.341146488 8 8.01 8.01 8.02 0.32504209 0.30898574

8.05 8.05 8.06 8.06 8.07 0.29020474 0.271462828.1 8.1 8.11 8.11 8.12 0.25093433 0.23043403

8.15 8.15 8.16 8.16 8.17 0.20917288 0.187927648.2 8.2 8.21 8.21 8.22 0.16698421 0.14604365

8.25 8.25 8.26 8.26 8.27 0.126452 0.106850038.3 8.3 8.31 8.31 8.32 0.08957681 0.07228061

8.35 8.35 8.36 8.36 8.37 0.05817739 0.044039678.4 8.4 8.41 8.41 8.42 0.03380095 0.02351791

8.45 8.45 8.46 8.46 8.47 0.01764692 0.0117248.5 8.5 8.51 8.51 8.52 0.01050798 0.00923497

8.55 8.55 8.56 8.56 8.57 0.01273109 0.016167998.6 8.6 8.61 8.61 8.62 0.02420069 0.03217487

8.65 8.65 8.66 8.66 8.67 0.04434468 0.056459558.7 8.7 8.71 8.71 8.72 0.07216294 0.08781768

8.75 8.75 8.76 8.76 8.77 0.10627697 0.124696328.8 8.8 8.81 8.81 8.82 0.14499826 0.16527092

8.85 8.85 8.86 8.86 8.87 0.1864118 0.207535498.9 8.9 8.91 8.91 8.92 0.22847079 0.24940185

8.95 8.95 8.96 8.96 8.97 0.26909786 0.288802779 9 9.01 9.01 9.02 0.30628761 0.32379404

9.05 9.05 9.06 9.06 9.07 0.33820563 0.352650429.1 9.1 9.11 9.11 9.12 0.36327896 0.37395068

9.15 9.15 9.16 9.16 9.17 0.38027364 0.386647629.2 9.2 9.21 9.21 9.22 0.38835542 0.39011954

9.25 9.25 9.26 9.26 9.27 0.38713068 0.384200689.3 9.3 9.31 9.31 9.32 0.37666567 0.36918913

9.35 9.35 9.36 9.36 9.37 0.35748286 0.34583189.4 9.4 9.41 9.41 9.42 0.33053495 0.31528731

9.45 9.45 9.46 9.46 9.47 0.29715748 0.279068459.5 9.5 9.51 9.51 9.52 0.25900271 0.23896733

9.55 9.55 9.56 9.56 9.57 0.21795777 0.196966649.6 9.6 9.61 9.61 9.62 0.17605135 0.15514167

9.65 9.65 9.66 9.66 9.67 0.13535345 0.115557749.7 9.7 9.71 9.71 9.72 0.09787309 0.08016824

9.75 9.75 9.76 9.76 9.77 0.06545919 0.050718219.8 9.8 9.81 9.81 9.82 0.03970929 0.02865832

9.85 9.85 9.86 9.86 9.87 0.02189089 0.015073379.9 9.9 9.91 9.91 9.92 0.01287901 0.010629

9.95 9.95 9.96 9.96 9.97 0.01311326 0.0155390510 10 10.01 10.01 10.02 0.02257636 0.02955527

10.05 10.05 10.06 10.06 10.07 0.04079529 0.05197987

Page 5: Ruge Kutta

10.1 10.1 10.11 10.11 10.12 0.06686496 0.0817002910.15 10.15 10.16 10.16 10.17 0.09949312 0.1172443410.2 10.2 10.21 10.21 10.22 0.13706445 0.15685316

10.25 10.25 10.26 10.26 10.27 0.17772053 0.1985682410.3 10.3 10.31 10.31 10.32 0.21945174 0.24032828

10.35 10.35 10.36 10.36 10.37 0.26019662 0.2800710310.4 10.4 10.41 10.41 10.42 0.29794366 0.31583511

10.45 10.45 10.46 10.46 10.47 0.33083068 0.3458568610.5 10.5 10.51 10.51 10.52 0.35723661 0.36865725

10.55 10.55 10.56 10.56 10.57 0.37586146 0.383114810.6 10.6 10.61 10.61 10.62 0.38579028 0.38852071

10.65 10.65 10.66 10.66 10.67 0.38653813 0.384613610.7 10.7 10.71 10.71 10.72 0.37807373 0.37159216

10.75 10.75 10.76 10.76 10.77 0.36082078 0.3501050510.8 10.8 10.81 10.81 10.82 0.3356367 0.3212186

10.85 10.85 10.86 10.86 10.87 0.30377003 0.2863638610.9 10.9 10.91 10.91 10.92 0.26679858 0.24726574

10.95 10.95 10.96 10.96 10.97 0.22655125 0.2058576311 11 11.01 11.01 11.02 0.18501758 0.16418582

11.05 11.05 11.06 11.06 11.07 0.14424941 0.1243083311.1 11.1 11.11 11.11 11.12 0.10625953 0.08819332

11.15 11.15 11.16 11.16 11.17 0.07292227 0.0576219311.2 11.2 11.21 11.21 11.22 0.04588104 0.03410043

11.25 11.25 11.26 11.26 11.27 0.02646732 0.0187860411.3 11.3 11.31 11.31 11.32 0.01563512 0.01242996

11.35 11.35 11.36 11.36 11.37 0.01391406 0.0153405411.4 11.4 11.41 11.41 11.42 0.0213835 0.02736832

11.45 11.45 11.46 11.46 11.47 0.03766893 0.0479137211.5 11.5 11.51 11.51 11.52 0.06196067 0.07595688

11.55 11.55 11.56 11.56 11.57 0.09305418 0.1101083411.6 11.6 11.61 11.61 11.62 0.12940979 0.14867782

11.65 11.65 11.66 11.66 11.67 0.16922891 0.1897579311.7 11.7 11.71 11.71 11.72 0.21054301 0.23131847

11.75 11.75 11.76 11.76 11.77 0.25131097 0.2713067611.8 11.8 11.81 11.81 11.82 0.28951989 0.30774905

11.85 11.85 11.86 11.86 11.87 0.32328448 0.3388478911.9 11.9 11.91 11.91 11.92 0.3509401 0.36307084

11.95 11.95 11.96 11.96 11.97 0.3711248 0.3792259412 12 12.01 12.01 12.02 0.38284643 0.38652039

12.05 12.05 12.06 12.06 12.07 0.38553134 0.3845994412.1 12.1 12.11 12.11 12.12 0.37905254 0.37356363

12.15 12.15 12.16 12.16 12.17 0.36373562 0.3539635712.2 12.2 12.21 12.21 12.22 0.34034243 0.32677246

12.25 12.25 12.26 12.26 12.27 0.31003318 0.2933377512.3 12.3 12.31 12.31 12.32 0.27430888 0.25531439

12.35 12.35 12.36 12.36 12.37 0.23493705 0.2145829212.4 12.4 12.41 12.41 12.42 0.19386422 0.17315645

12.45 12.45 12.46 12.46 12.47 0.15311976 0.1330811512.5 12.5 12.51 12.51 12.52 0.11471551 0.09633522

12.55 12.55 12.56 12.56 12.57 0.08054656 0.0647312412.6 12.6 12.61 12.61 12.62 0.05229764 0.03982671

Page 6: Ruge Kutta

12.65 12.65 12.66 12.66 12.67 0.03136011 0.0228473612.7 12.7 12.71 12.71 12.72 0.01876342 0.01462678

12.75 12.75 12.76 12.76 12.77 0.01512448 0.0155655312.8 12.8 12.81 12.81 12.82 0.02061745 0.0256116

12.85 12.85 12.86 12.86 12.87 0.03496545 0.0442632512.9 12.9 12.91 12.91 12.92 0.05745447 0.07059411

12.95 12.95 12.96 12.96 12.97 0.0869689 0.1032991413 13 13.01 13.01 13.02 0.12204691 0.14075935

13.05 13.05 13.06 13.06 13.07 0.16095281 0.1811219113.1 13.1 13.11 13.11 13.12 0.20176297 0.22239182

13.15 13.15 13.16 13.16 13.17 0.24246087 0.2625304813.2 13.2 13.21 13.21 13.22 0.28103682 0.29955643

13.25 13.25 13.26 13.26 13.27 0.31558713 0.3316431613.3 13.3 13.31 13.31 13.32 0.34440811 0.35720917

13.35 13.35 13.36 13.36 13.37 0.36608001 0.3749959813.4 13.4 13.41 13.41 13.42 0.37953704 0.38413001

13.45 13.45 13.46 13.46 13.47 0.38411973 0.3841655513.5 13.5 13.51 13.51 13.52 0.37960723 0.37510645

13.55 13.55 13.56 13.56 13.57 0.36622801 0.357405713.6 13.6 13.61 13.61 13.62 0.34464824 0.33194273

13.65 13.65 13.66 13.66 13.67 0.31593865 0.2999797813.7 13.7 13.71 13.71 13.72 0.28152141 0.26309927

13.75 13.75 13.76 13.76 13.77 0.24309965 0.223125513.8 13.8 13.81 13.81 13.82 0.20257322 0.1820344

13.85 13.85 13.86 13.86 13.87 0.16194473 0.1418558613.9 13.9 13.91 13.91 13.92 0.12322059 0.10457343

13.95 13.95 13.96 13.96 13.97 0.08831194 0.0720264514 14 14.01 14.01 14.02 0.0589403 0.04581925

14.05 14.05 14.06 14.06 14.07 0.03655271 0.027242114.1 14.1 14.11 14.11 14.12 0.02225043 0.01720769

14.15 14.15 14.16 14.16 14.17 0.01673475 0.0162062714.2 14.2 14.21 14.21 14.22 0.02027263 0.02428172

14.25 14.25 14.26 14.26 14.27 0.03268377 0.0410296514.3 14.3 14.31 14.31 14.32 0.05334979 0.06561764

14.35 14.35 14.36 14.36 14.37 0.08124506 0.0968266414.4 14.4 14.41 14.41 14.42 0.11498756 0.13311137

14.45 14.45 14.46 14.46 14.47 0.15290739 0.1726768514.5 14.5 14.51 14.51 14.52 0.19312939 0.21356721

14.55 14.55 14.56 14.56 14.57 0.23366585 0.2537623414.6 14.6 14.61 14.61 14.62 0.27251477 0.29127773

14.65 14.65 14.66 14.66 14.67 0.30775874 0.3242624114.7 14.7 14.71 14.71 14.72 0.33765952 0.35109031

14.75 14.75 14.76 14.76 14.77 0.36074384 0.3704403914.8 14.8 14.81 14.81 14.82 0.37587592 0.3813617

14.85 14.85 14.86 14.86 14.87 0.38231342 0.3833201114.9 14.9 14.91 14.91 14.92 0.3797438 0.37622446

14.95 14.95 14.96 14.96 14.97 0.36829951 0.3604307215 15 15.01 15.01 15.02 0.34855117 0.33672422

15.05 15.05 15.06 15.06 15.07 0.32147916 0.3062805215.1 15.1 15.11 15.11 15.12 0.28842486 0.27060717

15.15 15.15 15.16 15.16 15.17 0.2510243 0.23146907

Page 7: Ruge Kutta

15.2 15.2 15.21 15.21 15.22 0.2111271 0.1908010815.25 15.25 15.26 15.26 15.27 0.17070504 0.1506124715.3 15.3 15.31 15.31 15.32 0.13175459 0.11288755

15.35 15.35 15.36 15.36 15.37 0.09619833 0.0794877815.4 15.4 15.41 15.41 15.42 0.06579004 0.05205985

15.45 15.45 15.46 15.46 15.47 0.04202817 0.0319545515.5 15.5 15.51 15.51 15.52 0.02608207 0.02016022

15.55 15.55 15.56 15.56 15.57 0.01873436 0.0172541615.6 15.6 15.61 15.61 15.62 0.0203426 0.0233744

15.65 15.65 15.66 15.66 15.67 0.03082183 0.0382131315.7 15.7 15.71 15.71 15.72 0.04964909 0.06103219

15.75 15.75 15.76 15.76 15.77 0.07588948 0.0906997915.8 15.8 15.81 15.81 15.82 0.1082426 0.12574663

15.85 15.85 15.86 15.86 15.87 0.14510699 0.1644386815.9 15.9 15.91 15.91 15.92 0.1846594 0.20486297

15.95 15.95 15.96 15.96 15.97 0.22494496 0.2450221616 16 16.01 16.01 16.02 0.2639738 0.28293324

16.05 16.05 16.06 16.06 16.07 0.29981935 0.3167254516.1 16.1 16.11 16.11 16.12 0.33071342 0.34473258

16.15 16.15 16.16 16.16 16.17 0.35513342 0.3655751116.2 16.2 16.21 16.21 16.22 0.37187743 0.37822822

16.25 16.25 16.26 16.26 16.27 0.38012331 0.3820721116.3 16.3 16.31 16.31 16.32 0.3794691 0.37692239

16.35 16.35 16.36 16.36 16.37 0.36995262 0.3630389316.4 16.4 16.41 16.41 16.42 0.35204926 0.34111271

16.45 16.45 16.46 16.46 16.47 0.32664834 0.3122314816.5 16.5 16.51 16.51 16.52 0.29500884 0.27782577

16.55 16.55 16.56 16.56 16.57 0.25869706 0.2395980516.6 16.6 16.61 16.61 16.62 0.21950908 0.19943845

16.65 16.65 16.66 16.66 16.67 0.17938186 0.1593313716.7 16.7 16.71 16.71 16.72 0.14029759 0.12125737

16.75 16.75 16.76 16.76 16.77 0.10418572 0.0870954116.8 16.8 16.81 16.81 16.82 0.07282771 0.05853007

16.85 16.85 16.86 16.86 16.87 0.04776918 0.0369685516.9 16.9 16.91 16.91 16.92 0.03024371 0.02347129

16.95 16.95 16.96 16.96 16.97 0.02111209 0.0186998417 17 17.01 17.01 17.02 0.0208201 0.02288445

17.05 17.05 17.06 17.06 17.07 0.02937669 0.0358129417.1 17.1 17.11 17.11 17.12 0.04635387 0.05684147

17.15 17.15 17.16 17.16 17.17 0.07090804 0.0849265917.2 17.2 17.21 17.21 17.22 0.10182198 0.11867703

17.25 17.25 17.26 17.26 17.27 0.13756515 0.1564225917.3 17.3 17.31 17.31 17.32 0.17636947 0.19629684

17.35 17.35 17.36 17.36 17.37 0.21631678 0.2363293417.4 17.4 17.41 17.41 17.42 0.25543367 0.27454309

17.45 17.45 17.46 17.46 17.47 0.29178894 0.3090521117.5 17.5 17.51 17.51 17.52 0.32358898 0.33815453

17.55 17.55 17.56 17.56 17.57 0.34926622 0.3604165317.6 17.6 17.61 17.61 17.62 0.36755643 0.37474297

17.65 17.65 17.66 17.66 17.67 0.37756094 0.3804312717.7 17.7 17.71 17.71 17.72 0.37879082 0.37720586

Page 8: Ruge Kutta

17.75 17.75 17.76 17.76 17.77 0.37119075 0.3652315117.8 17.8 17.81 17.81 17.82 0.35514148 0.34510495

17.85 17.85 17.86 17.86 17.87 0.3314408 0.3178251317.9 17.9 17.91 17.91 17.92 0.30126383 0.28474361

17.95 17.95 17.96 17.96 17.97 0.26610479 0.2474976218 18 18.01 18.01 18.02 0.22770301 0.20792906

18.05 18.05 18.06 18.06 18.07 0.18795688 0.1679933618.1 18.1 18.11 18.11 18.12 0.14883001 0.12966289

18.15 18.15 18.16 18.16 18.17 0.1122542 0.0948295218.2 18.2 18.21 18.21 18.22 0.0800341 0.06521126

18.25 18.25 18.26 18.26 18.27 0.05375813 0.0422675218.3 18.3 18.31 18.31 18.32 0.03472022 0.02712722

18.35 18.35 18.36 18.36 18.37 0.02385604 0.0205332118.4 18.4 18.41 18.41 18.42 0.02169705 0.02280585

18.45 18.45 18.46 18.46 18.47 0.0283445 0.0338274318.5 18.5 18.51 18.51 18.52 0.04346467 0.05304827

18.55 18.55 18.56 18.56 18.57 0.06630566 0.0795141418.6 18.6 18.61 18.61 18.62 0.09573476 0.11191361

18.65 18.65 18.66 18.66 18.67 0.13029461 0.1486430318.7 18.7 18.71 18.71 18.72 0.16827538 0.18788594

18.75 18.75 18.76 18.76 18.77 0.20779935 0.2277028518.8 18.8 18.81 18.81 18.82 0.24691378 0.26612714

18.85 18.85 18.86 18.86 18.87 0.28368733 0.3012621818.9 18.9 18.91 18.91 18.92 0.31630547 0.33137492

18.95 18.95 18.96 18.96 18.97 0.34315999 0.3549814119 19 19.01 19.01 19.02 0.36292829 0.37091991

19.05 19.05 19.06 19.06 19.07 0.37463853 0.3784080719.1 19.1 19.11 19.11 19.12 0.37771741 0.37708132

19.15 19.15 19.16 19.16 19.17 0.37201819 0.3670106119.2 19.2 19.21 19.21 19.22 0.35782776 0.34869866

19.25 19.25 19.26 19.26 19.27 0.3358521 0.3230548619.3 19.3 19.31 19.31 19.32 0.30718125 0.29135009

19.35 19.35 19.36 19.36 19.37 0.27323518 0.2551537619.4 19.4 19.41 19.41 19.42 0.2356935 0.21625612

19.45 19.45 19.46 19.46 19.47 0.19641235 0.1765797219.5 19.5 19.51 19.51 19.52 0.15733261 0.13808437

19.55 19.55 19.56 19.56 19.57 0.12038402 0.1026703419.6 19.6 19.61 19.61 19.62 0.0873899 0.07208458

19.65 19.65 19.66 19.66 19.67 0.05997715 0.0478345319.7 19.7 19.71 19.71 19.72 0.03949604 0.03111379

19.75 19.75 19.76 19.76 19.77 0.0269537 0.0227434619.8 19.8 19.81 19.81 19.82 0.02296462 0.02313171

19.85 19.85 19.86 19.86 19.87 0.02772051 0.0322539819.9 19.9 19.91 19.91 19.92 0.04098111 0.04965439

19.95 19.95 19.96 19.96 19.97 0.06208632 0.0744685520 20 20.01 20.01 20.02 0.08998907 0.10546649

Page 9: Ruge Kutta

0.02 N= 1000 4 5000 100

X3 X4 Y1 Y2 Y3 Y4 F1 F20.0025 0.005 0 0.25 0.25 0.49375 25 25

0.012375 0.01962656 0.49375 0.7375 0.73132813 0.9628125 24.375 23.75781250.03150936 0.04315145 0.9628125 1.1882793 1.17624414 1.38403911 22.5466797 21.34316410.05895244 0.07440743 1.38403911 1.5800998 1.56279932 1.736658 19.6060693 17.87602040.09334392 0.11184624 1.736658 1.89364871 1.87194049 2.00329821 15.699071 13.52824850.13298115 0.15361522 2.00329821 2.1134904 2.08844918 2.17084534 11.0192198 8.515097070.17590349 0.19764904 2.17084534 2.22882631 2.20169074 2.23108662 5.79809688 3.084540210.21998929 0.24177177 2.23108662 2.23402532 2.20613674 2.18111339 0.29387002 -2.494988260.26306076 0.28380447 2.18111339 2.12889867 2.10163475 2.02346149 -5.22147183 -7.947863560.30299153 0.32167272 2.02346149 1.9187059 1.89341263 1.76598266 -10.4755587 -13.00488560.33781164 0.35350906 1.76598266 1.61389176 1.59181698 1.42145356 -15.2090903 -17.41656860.36580473 0.37774504 1.42145356 1.22956724 1.21179907 1.00694173 -19.1886327 -20.96544970.38559265 0.39318852 1.00694173 0.78476043 0.77217365 0.54296011 -22.2181304 -23.47680760.39620326 0.39908227 0.54296011 0.30147447 0.29468746 0.05245196 -24.1485645 -24.82726470.39711826 0.39514114 0.05245196 -0.19640087 -0.19705652 -0.44034368 -24.8852832 -24.95084810.38829843 0.38156582 -0.44034368 -0.6842701 -0.67876581 -0.91108977 -24.3926419 -23.84221230.37018535 0.35903265 -0.91108977 -1.13804704 -1.12665842 -1.33655314 -22.6957274 -21.55686520.34367921 0.32865991 -1.33655314 -1.53534395 -1.51863704 -1.69575117 -19.8790813 -18.20838990.31009415 0.29195233 -1.69575117 -1.85657606 -1.83537917 -1.97098654 -16.0824887 -13.96279980.27109306 0.25072653 -1.97098654 -2.08592695 -2.06128962 -2.14871918 -11.4940408 -9.030307630.22860526 0.20702117 -2.14871918 -2.21212735 -2.18526836 -2.22023234 -6.34081676 -3.654917780.18473108 0.16299605 -2.22023234 -2.22900879 -2.20125589 -2.18206003 -0.87764586 1.89764456

0.141638 0.12082546 -2.18206003 -2.13580509 -2.10852934 -2.03615503 4.62549386 7.35306890.10145359 0.08259076 -2.03615503 -1.93718686 -1.91173492 -1.78978901 9.89681722 12.4420110.06616049 0.05017766 -1.78978901 -1.64302747 -1.6206551 -1.45519023 14.6761545 16.91339080.03749854 0.02518321 -1.45519023 -1.26791231 -1.24972243 -1.04893658 18.7277923 20.54678010.01687906 0.00883714 -1.04893658 -0.8304156 -0.81730389 -0.59113422 21.8520984 23.16326910.00531533 0.00194131 -0.59113422 -0.35218064 -0.34479147 -0.10442255 23.8953581 24.63427590.00337281 0.00483043 -0.10442255 0.14315082 0.1444561 0.38714542 24.7573368 24.88786490.0111415 0.01735579 0.38714542 0.63110738 0.62626807 0.85929166 24.3961965 23.9122647

0.02823176 0.0388929 0.85929166 1.08759693 1.07685578 1.28871227 22.8305263 21.75641180.05379387 0.06837265 1.28871227 1.49009614 1.47398723 1.6542276 20.1383869 18.52749650.08656027 0.10433433 1.6542276 1.81876179 1.79808394 1.93782693 16.4534188 14.38563430.12490842 0.14499805 1.93782693 2.05740902 2.03318619 2.12555589 11.9582089 9.535925270.16694114 0.18835283 2.12555589 2.19430832 2.16773887 2.20820305 6.87524348 4.218298620.21058045 0.23225602 2.20820305 2.22276201 2.19515947 2.18175192 1.4558963 -1.304357510.25367034 0.27453922 2.18175192 2.1414319 2.11416 2.04757608 -4.03200238 -6.759192290.29408324 0.31311537 2.04757608 1.95440205 1.92880735 1.81236798 -9.31740238 -11.87687250.3298251 0.34608175 1.81236798 1.67097377 1.64831917 1.48780522 -14.1394208 -16.4048807

0.35913378 0.37181386 1.48780522 1.30520303 1.28660547 1.08997077 -18.2602187 -20.11997520.38056589 0.38904544 1.08997077 0.87520345 0.86157881 0.63855604 -21.4767324 -22.83919580.39306793 0.39693078 0.63855604 0.40224924 0.39426729 0.15588622 -23.6306794 -24.42887440.39602801 0.39508626 0.15588622 -0.09027726 -0.09222584 -0.3341838 -24.6163476 -24.81120540.38930585 0.38360898 -0.3341838 -0.57804163 -0.57386434 -0.80744842 -24.385783 -23.96805330.37323938 0.36307165 -0.80744842 -1.03695965 -1.02686654 -1.24054688 -22.9511222 -21.94181170.34862778 0.33449405 -1.24054688 -1.44438643 -1.4288796 -1.61211633 -20.3839558 -18.83327220.31669172 0.29929241 -1.61211633 -1.7802339 -1.76008245 -1.90384562 -16.8117568 -14.79661140.27901279 0.25920914 -1.90384562 -2.02796113 -2.00416306 -2.10137761 -12.4115507 -10.0317437

𝑥 ̈=𝑥 ̇= 𝑥=𝐹(𝑡)=

0 1 2 3 4 5

-1

0

1

POSICION Y VELOCIDAD VS TIEMPO

POSICIÓN VELOCIDAD

Page 10: Ruge Kutta

0.23745525 0.21622671 -2.10137761 -2.17538904 -2.14912182 -2.19501574 -7.40114308 -4.774421070.19407372 0.17246948 -2.19501574 -2.21529913 -2.18786143 -2.18020004 -2.02833853 0.715431140.15101161 0.13009879 -2.18020004 -2.14578689 -2.11853438 -2.05772906 3.44131504 6.166565090.11039527 0.09120617 -2.05772906 -1.97035255 -1.94463094 -1.83371723 8.73765101 11.30981230.07422892 0.0577101 -1.83371723 -1.69772495 -1.67480348 -1.51928954 13.5992284 15.89137490.04429583 0.03126138 -1.51928954 -1.34142717 -1.32243605 -1.13002912 17.7862371 19.6853490.02207032 0.01316177 -1.13002912 -0.91910585 -0.90498048 -0.68520493 21.0923272 22.50486360.0086452 0.00429993 -0.68520493 -0.45165714 -0.44309208 -0.20681793 23.3547784 24.21128450.004678 0.00510778 -0.20681793 0.03780715 0.04039238 0.28148706 24.4625086 24.721031

0.0103588 0.01553945 0.28148706 0.52510234 0.52158375 0.75559006 24.3615276 24.00966880.02540111 0.03507387 0.75559006 0.98616574 0.97672086 1.19208728 23.0575683 22.11308070.04905632 0.06274075 1.19208728 1.39824494 1.38334385 1.56944648 20.6157661 19.1256570.08015095 0.0971688 1.56944648 1.74102054 1.72140246 1.86906909 17.1574065 15.19559840.11714488 0.13665369 1.86906909 1.9976081 1.97424473 2.0762069 12.8539003 10.5175640.15820759 0.17924244 2.0762069 2.15538978 2.1294372 2.18068792 7.91828851 5.323029890.20130878 0.22282996 2.18068792 2.20663488 2.17937628 2.17741596 2.59469552 -0.131164380.24431875 0.26526318 2.17741596 2.14887851 2.12166081 2.06661909 -2.85374517 -5.575515120.28511358 0.30444733 2.06661909 1.98504012 1.95920738 1.85383515 -8.1578972 -10.74117110.32168009 0.33844939 1.85383515 1.72327599 1.70010305 1.54963493 -13.0559157 -15.37320960.35221512 0.36559344 1.54963493 1.3765732 1.35720277 1.16909714 -17.3061733 -19.2432170.37521449 0.38454327 1.16909714 0.96210534 0.94749162 0.7310609 -20.6991802 -22.16055160.38954709 0.39436815 0.7310609 0.50038181 0.49124355 0.25719317 -23.0679093 -23.98173540.39451047 0.39458851 0.25719317 0.01423299 0.01101808 -0.22908302 -24.2960181 -24.61750960.38986532 0.3851994 -0.22908302 -0.47231865 -0.46945511 -0.70374632 -24.3235633 -24.03720950.37584695 0.36667043 -0.70374632 -0.93524557 -0.92644874 -1.14336369 -23.1499255 -22.27024260.35315341 0.33992224 -1.14336369 -1.35170172 -1.33740968 -1.52624722 -20.8338037 -19.40459910.32291074 0.3062808 -1.52624722 -1.70115001 -1.68207192 -1.83352406 -17.4902795 -15.58247050.28661705 0.26741168 -1.83352406 -1.96637505 -1.943456 -2.05006667 -13.2850997 -10.99319460.24606836 0.22523757 -2.05006667 -2.13433127 -2.10870544 -2.16523759 -8.42645967 -5.863876330.20326972 0.18184319 -2.16523759 -2.19678455 -2.16971908 -2.17341189 -3.15469607 -0.448149090.16033603 0.13937222 -2.17341189 -2.15071588 -2.12354823 -2.07425196 2.26960162 4.986366490.11938755 0.09992144 -2.07425196 -1.99846724 -1.97253909 -1.87272084 7.57847219 10.17128710.08244521 0.06543717 -1.87272084 -1.74762264 -1.72421363 -1.57883388 12.5098199 14.8507210.05133086 0.03761927 -1.57883388 -1.41063034 -1.39089491 -1.20716104 16.820354 18.79389630.02757742 0.01783736 -1.20716104 -1.00418512 -0.98909561 -0.77610458 20.2975913 21.80654260.01235334 0.00706336 -0.77610458 -0.54840128 -0.53869997 -0.30698795 22.7703303 23.7404610.00640519 0.00582376 -0.30698795 -0.06581714 -0.06197979 0.17699909 24.1170802 24.50081520.01002096 0.0141739 0.17699909 0.41971939 0.4175069 0.6519467 24.2720297 24.05078090.02301619 0.0316955 0.6519467 0.88422932 0.87607999 1.09440621 23.2282625 22.41332910.04474337 0.05751764 1.09440621 1.30478684 1.29110676 1.48254779 21.0380625 19.67005470.07412414 0.09036001 1.48254779 1.66065075 1.6421189 1.79723744 17.8102956 15.95711090.10970289 0.12859645 1.79723744 1.93428742 1.91182195 2.02298022 13.7049983 11.45845160.1497188 0.1703354 2.02298022 2.11223465 2.0869474 2.14868322 8.92544347 6.39671819

0.19219304 0.21351351 2.14868322 2.18576397 2.15890543 2.16820062 3.70807496 1.022220940.2350266 0.25599763 2.16820062 2.15130874 2.12420623 2.08063413 -1.68918861 -4.399439390.276104 0.29569022 2.08063413 2.01063709 1.98462916 1.89037412 -6.99970418 -9.60049685

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0.19066199 0.17070504 -2.03260186 -2.04651073 -2.02110321 -2.00925684 -1.39088777 1.149864550.15097866 0.13175459 -2.00925684 -1.97263814 -1.94752243 -1.88670349 3.66187026 6.173441320.11374062 0.09619833 -1.88670349 -1.80139672 -1.77781293 -1.67105504 8.53067634 10.88905570.0807853 0.06579004 -1.67105504 -1.54130295 -1.52041477 -1.37301829 12.9752087 15.0640275

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F3 F4 POSICIÓN VELOCIDAD 0.0524.6875 24.375 0.0049791667 0.49583333 0.1

23.453125 22.5466797 0.0196473958 0.47614518 0.1521.0613306 19.6060693 0.0432128908 0.44283236 0.217.6309446 15.699071 0.0745064298 0.39754835 0.2513.3320102 11.0192198 0.1119778803 0.34253679 0.38.37735682 5.79809688 0.1537729838 0.28052032 0.353.01206399 0.29387002 0.1978251119 0.21456581 0.4-2.49866163 -5.22147183 0.2419574536 0.1479327 0.45-7.88259517 -10.4755587 0.2839905804 0.08391195 0.5-12.8739411 -15.2090903 0.3218500737 0.02566346 0.55-17.226455 -19.1886327 0.3536689013 -0.02393985 0.6

-20.7255918 -22.2181304 0.3778794882 -0.06245376 0.65-23.1990809 -24.1485645 0.3932909432 -0.0879832 0.7-24.5254076 -24.8852832 0.3991476361 -0.09927546 0.75-24.6397821 -24.3926419 0.3951662438 -0.09578169 0.8-23.5373043 -22.6957274 0.3815494509 -0.07768354 0.85-21.2731686 -19.8790813 0.3589756395 -0.04588377 0.9-17.9599014 -16.0824887 0.3285650962 -0.00196119 0.95-13.7617687 -11.4940408 0.2918244159 0.05190769 1-8.88663212 -6.34081676 0.2505718636 0.11305654 1.05-3.57565757 -0.87764586 0.206847391 0.17846116 1.11.90861513 4.62549386 0.1628117611 0.24488906 1.157.29525023 9.89681722 0.1206397693 0.30905921 1.212.3183008 14.6761545 0.0824128369 0.36780411 1.2516.7299389 18.7277923 0.0500162826 0.41822612 1.320.3126827 21.8520984 0.0250463407 0.45784038 1.3522.8901179 23.8953581 0.0087315138 0.48469731 1.424.3355839 24.7573368 0.001872132 0.49747852 1.4524.5783982 24.3961965 0.004801095 0.4955615 1.523.6073123 22.8305263 0.0173677213 0.47905002 1.5521.4710302 20.1383869 0.0389454871 0.44876845 1.618.2757667 16.4534188 0.0684632603 0.40622068 1.6514.1799666 11.9582089 0.1044584696 0.35351541 1.79.38644766 6.87524348 0.1451495726 0.29326166 1.754.13235808 1.4558963 0.1885242299 0.22843956 1.8-1.32255622 -4.03200238 0.2324388227 0.16225305 1.85-6.70879226 -9.31740238 0.2747243917 0.09797144 1.9-11.760405 -14.1394208 0.313293762 0.03876796 1.95-16.228138 -18.2602187 0.3462445631 -0.01243689 2

-19.8917225 -21.4767324 0.3719530595 -0.05311941 2.05-22.5707367 -23.6306794 0.3891541633 -0.08127706 2.1-24.1334909 -24.6163476 0.3970036862 -0.09552709 2.15-24.503501 -24.385783 0.3951197684 -0.09517436 2.2-23.663231 -22.9511222 0.3836014503 -0.08024531 2.25

-21.6549226 -20.3839558 0.3630234853 -0.05148618 2.3-18.5784728 -16.8117568 0.3344076624 -0.0103257 2.35-14.5864645 -12.4115507 0.2991720726 0.04119586 2.4-9.87659932 -7.40114308 0.2590608338 0.10052781 2.45

Page 18: Ruge Kutta

-4.68190678 -2.02833853 0.2160577611 0.16473535 2.50.74078538 3.44131504 0.172288252 0.23064474 2.556.12354865 8.73765101 0.1299142409 0.29500012 2.611.2005917 13.5992284 0.0910274144 0.35462442 2.6515.7213845 17.7862371 0.0575459611 0.40657612 2.719.463021 21.0923272 0.0311199533 0.44829445 2.75

22.2412095 23.3547784 0.0130500269 0.47772555 2.823.9193498 24.4625086 0.0042233686 0.49342363 2.8524.4152496 24.3615276 0.0050701586 0.49462192 2.923.7051497 23.0575683 0.0155426098 0.48127022 2.9521.8248611 20.6157661 0.0351176223 0.45403685 318.8679599 17.1574065 0.0628229089 0.41427523 3.0514.9811308 12.8539003 0.0972852866 0.36395663 3.110.3568902 7.91828851 0.1367987361 0.30557245 3.155.22405129 2.59469552 0.1794088545 0.24201085 3.2-0.16359807 -2.85374517 0.2230095231 0.17641385 3.25-5.53984331 -8.1578972 0.265447007 0.1120221 3.3-10.6391974 -13.0559157 0.3046263418 0.05201468 3.35-15.2100106 -17.3061733 0.3386147525 -0.00064777 3.4-19.0268898 -20.6991802 0.3657369997 -0.04336917 3.45-21.9018119 -23.0679093 0.3846579481 -0.07404595 3.5-23.6933865 -24.2960181 0.39444829 -0.09117063 3.55-24.3138094 -24.3235633 0.394630186 -0.09390592 3.6-23.733165 -23.1499255 0.3852005837 -0.08212562 3.65

-21.9808686 -20.8338037 0.3666310755 -0.05642043 3.7-19.1441765 -17.4902795 0.3398443171 -0.01806835 3.75-15.363842 -13.2850997 0.3061681854 0.03102889 3.8

-10.8271309 -8.42645967 0.2672699546 0.0884401 3.85-5.75854558 -3.15469607 0.2250737518 0.15132501 3.9-0.40871539 2.26960162 0.1816653795 0.21657482 3.954.95799647 7.57847219 0.1391892135 0.28096585 410.0765562 12.5098199 0.0997422709 0.34131882 4.0514.6943482 16.820354 0.0652706831 0.39465582 4.118.5836419 20.2975913 0.0374736833 0.43834724 4.1521.5528227 22.7703303 0.0177198461 0.47024144 4.223.4558319 24.1170802 0.0069797074 0.48877068 4.2524.1993517 24.2720297 0.0057780823 0.49302817 4.323.7473805 23.2282625 0.0141684234 0.48281248 4.3522.1229758 21.0380625 0.0317304719 0.45863702 4.419.407079 17.8102956 0.057591304 0.42170425 4.45

15.7344822 13.7049983 0.090468717 0.3738459 4.511.2871391 8.92544347 0.1287348012 0.31743209 4.556.28515015 3.70807496 0.1704965441 0.25525397 4.6

0.97587 -1.68918861 0.2136894758 0.19038563 4.65-4.37832453 -6.99970418 0.2561797245 0.12603211 4.7-9.51300054 -11.9612771 0.295869436 0.06537105 4.75-14.1747288 -16.3291063 0.3308003491 0.01139578 4.8-18.1335919 -19.8878625 0.3592504141 -0.03323236 4.85-21.1945238 -22.4623037 0.3798186868 -0.06631529 4.9-23.2069214 -23.9259043 0.3914943125 -0.08622641 4.95-24.072054 -24.2070731 0.3937062045 -0.09199057 5

Page 19: Ruge Kutta

-23.7479068 -23.292655 0.3863509741 -0.08333186 5.05-22.2512213 -21.2285445 0.369797748 -0.06068676 5.1-19.6566316 -18.1173827 0.3448696481 -0.0251822 5.15-16.0929435 -14.1134537 0.3128028628 0.02142058 5.2-11.7367401 -9.4150339 0.275185346 0.07681335 5.25-6.80363204 -4.25457304 0.2338781825 0.13825523 5.3-1.53758614 1.11280564 0.1909235134 0.20270833 5.353.80113996 6.42191816 0.1484435777 0.2669879 5.48.94886008 11.4106216 0.1085358613 0.32791975 5.4513.6514832 15.8327575 0.0731695318 0.38249695 5.517.6770553 19.470298 0.0440882752 0.42802826 5.5520.8272002 22.1440953 0.0227243267 0.46227079 5.622.9468951 23.7227048 0.0101279351 0.48354051 5.6523.9320995 24.128846 0.0069157343 0.49079508 5.723.7348615 23.3431863 0.0132405602 0.4836849 5.7522.365651 21.4052593 0.0287841889 0.46256994 5.8

19.8928068 18.4114765 0.0527733426 0.42850157 5.8516.4391261 14.5103309 0.0840181569 0.38317018 5.912.1757667 9.89503211 0.1209711968 0.32882131 5.957.31376496 4.79393729 0.1618040944 0.26814443 62.0935897 -0.5407473 0.2044980117 0.20413983 6.05

-3.22675108 -5.84543577 0.2469434544 0.13997029 6.1-8.38446217 -10.8581862 0.2870445004 0.0788048 6.15-13.1249417 -15.331635 0.3228222944 0.02366201 6.2-17.2143488 -19.0452037 0.3525126964 -0.02273867 6.25-20.4511391 -21.8159746 0.3746532679 -0.05811125 6.3-22.675997 -23.5077013 0.388155305 -0.08071575 6.35

-23.7796771 -24.0375074 0.3923573707 -0.08944376 6.4-23.7083688 -23.3799469 0.3870576949 -0.08387283 6.45-22.4663185 -21.5682246 0.372523855 -0.06428691 6.5-20.1155848 -18.6925209 0.349479272 -0.03166184 6.55-16.7729378 -14.8955029 0.3190672049 0.01238321 6.6-12.604059 -10.3652464 0.2827940343 0.06566611 6.65

-7.81532932 -5.32592099 0.2424546489 0.12554992 6.7-2.64361221 -0.0266968 0.2000436304 0.18907332 6.752.65546198 5.27057513 0.1576566328 0.2530971 6.87.82013147 10.3043018 0.1173868306 0.31445929 6.8512.5954333 14.8260664 0.081221553 0.37013128 6.9

16.74579 18.6128873 0.05094421 0.41736735 6.9520.0666309 21.4782143 0.0280463479 0.45384006 722.3944755 23.2811182 0.013654173 0.47775492 7.0523.6149813 23.9332223 0.0084731618 0.4879387 7.123.6685595 23.4030342 0.0127534823 0.48389697 7.1522.5532847 21.7174655 0.0262779206 0.46583811 7.220.3249536 18.9604677 0.0483728967 0.43466256 7.2517.0942944 15.2688502 0.0779420089 0.39191795 7.313.0214648 10.8254924 0.1135204382 0.33972227 7.358.30811231 5.85028373 0.153347513 0.28065894 7.43.18739091 0.5892422 0.1954538381 0.21764889 7.45-2.08757236 -4.69765071 0.2377586779 0.15380617 7.5-7.2561898 -9.74929741 0.278172778 0.09228404 7.55

Page 20: Ruge Kutta

-12.0632862 -14.3163786 0.3147015433 0.0361192 7.6-16.2716967 -18.173658 0.3455434788 -0.01191808 7.65-19.6739676 -21.1310903 0.3691790327 -0.0494606 7.7-22.1025829 -23.0431847 0.3844454625 -0.07466089 7.75-23.4382116 -23.8161618 0.3905940344 -0.0862821 7.8-23.6155711 -23.4125526 0.3873267421 -0.08375872 7.85-22.6266178 -21.8530149 0.3748107448 -0.06722409 7.9-20.5209089 -19.2152764 0.3536698261 -0.03750339 7.95-17.4031194 -15.6302615 0.3249533071 0.00392755 8-13.4278392 -11.2755929 0.2900839658 0.05501553 8.05-8.79190798 -6.36679152 0.2507875458 0.11323172 8.1-3.72466888 -1.14660968 0.2090073473 0.17569684 8.151.52337741 4.12697314 0.1668081268 0.23932357 8.26.69295601 9.19349963 0.1262740592 0.30096912 8.2511.528827 13.8028982 0.089405803 0.35759048 8.3

15.7923875 17.7278261 0.058021754 0.40639464 8.3519.273444 20.7748816 0.0336683599 0.44497632 8.4

21.8005752 22.7941344 0.0175439206 0.47143658 8.4523.249573 23.6865025 0.0104396274 0.4844762 8.5

23.5495472 23.4086137 0.0127007433 0.48345954 8.5522.6863933 21.9749132 0.0242098278 0.46844548 8.620.7034543 19.4569146 0.044392822 0.44018409 8.6517.6993443 15.9796329 0.0722476797 0.40007917 8.713.823045 11.7153781 0.1063941146 0.35011854 8.75

9.26651729 6.87521689 0.1451419966 0.29277547 8.84.25519511 1.69852537 0.1865750064 0.23088628 8.85-0.96316764 -3.55884912 0.2286454083 0.16751003 8.9-6.13074576 -8.63723284 0.2692752539 0.10577734 8.95-10.992381 -13.2859514 0.3064590165 0.04873573 9

-15.3081811 -17.2757032 0.338362584 -0.00080081 9.05-18.8653564 -20.4098697 0.3634137275 -0.04039068 9.1-21.488712 -22.5342051 0.3803795804 -0.06808494 9.15

-23.0492754 -23.544427 0.388427311 -0.08252332 9.2-23.4706376 -23.3913353 0.3871650042 -0.083001 9.25-22.7326937 -22.0832083 0.3766607422 -0.06950299 9.3-20.8726008 -19.6853576 0.3574389589 -0.0427045 9.35-17.9829083 -16.3168685 0.3304542605 -0.00393712 9.4-14.2069526 -12.1446856 0.2970440184 0.04487733 9.45-9.73174815 -7.37533894 0.2588620901 0.10132204 9.5-4.77872463 -2.24472089 0.2177969517 0.16260491 9.550.40722881 2.99358123 0.1758782947 0.22569663 9.65.56987143 8.08081888 0.1351767069 0.28748043 9.6510.4542718 12.7658632 0.0977013963 0.34490608 9.714.8193963 16.8176017 0.0653010076 0.39514042 9.7518.4500032 20.0363391 0.0395724312 0.43570716 9.821.1672562 22.2636391 0.0217820987 0.46460899 9.8522.8375336 23.3901234 0.0128036473 0.48042585 9.923.3789985 23.3608423 0.0130750197 0.48238472 9.9522.7656085 22.1779552 0.0225771059 0.47039744 1021.0283672 19.9005884 0.0408349691 0.44506456 10.0518.2537583 16.6418801 0.0669415806 0.40764535 10.1

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14.5794399 12.5633602 0.0996028783 0.35999512 10.1510.1874155 7.86694339 0.1372019102 0.30447328 10.25.29501856 2.78493342 0.1778788827 0.24382653 10.250.14415824 -2.43146785 0.2196231529 0.18105294 10.3-5.01064197 -7.5245769 0.2603726126 0.11925375 10.35-9.91482111 -12.2429578 0.2981155528 0.06148017 10.4-14.3263517 -16.3538348 0.3309899766 0.01058275 10.45-18.027684 -19.6545764 0.3573754572 -0.0309293 10.5

-20.8364742 -21.9826824 0.3759730104 -0.06101179 10.55-22.6145671 -23.223785 0.3858690432 -0.07818622 10.6-23.2747916 -23.3172657 0.3865802322 -0.08161243 10.65-22.785234 -22.2592165 0.3780771256 -0.07112971 10.7

-21.1707797 -20.1025977 0.3607853177 -0.04726431 10.75-18.5118491 -16.9545871 0.335564151 -0.01120304 10.8-14.9403923 -12.9712542 0.3036640113 0.03526578 10.85-10.6333413 -8.34982272 0.2666643396 0.08984096 10.9-5.8038442 -3.31890583 0.2263954311 0.14982241 10.95

-0.69071767 1.87280295 0.1848478954 0.21224476 114.45336269 6.96882319 0.1440742558 0.2740239 11.059.37434861 11.7175581 0.1060875518 0.3321094 11.113.8293657 15.8847163 0.0727619535 0.38363512 11.1517.5987002 19.2648705 0.045740296 0.42606062 11.220.496635 21.691585 0.0263530939 0.45729643 11.25

22.3805997 23.0456104 0.0155530299 0.47580693 11.323.1581847 23.260743 0.0138681418 0.48068589 11.3522.7916736 22.3270619 0.0213760082 0.47170078 11.421.2998716 20.291384 0.037700196 0.44930387 11.4518.7571429 17.2549168 0.0620291294 0.41460945 11.515.289703 13.3682274 0.093156437 0.36933843 11.55

11.0693546 8.82377613 0.1295407688 0.31573294 11.66.30497515 3.84638676 0.1693821197 0.25644532 11.651.23217877 -1.31787601 0.210710878 0.19440686 11.7-3.89833519 -6.41387105 0.2514851955 0.13268289 11.75-8.83317191 -11.1899859 0.2896918638 0.07432133 11.8-13.328756 -15.41056 0.3234457149 0.02220215 11.85

-17.1633541 -18.8675125 0.351082637 -0.0211047 11.9-20.1480108 -21.3906005 0.3712416164 -0.05346605 11.95-22.1358599 -22.8558033 0.3829317613 -0.07329056 12-23.0293512 -23.1914179 0.3855810063 -0.07960698 12.05-22.7850369 -22.381568 0.3790641042 -0.07211172 12.1-21.4156841 -20.4669531 0.3637085351 -0.05118345 12.15-18.9896098 -17.5428042 0.3402780555 -0.01786394 12.2-15.6272725 -13.754147 0.3099347119 0.02619373 12.25-11.4952917 -9.28860967 0.2741812106 0.07880725 12.3-6.79819132 -4.36713074 0.2347864991 0.13737314 12.35-1.7682761 0.76697191 0.1936982463 0.19899557 12.43.34585717 5.86003062 0.152946551 0.26062958 12.458.29160622 10.6605612 0.1145436388 0.31923148 12.5

12.82484 14.9316804 0.0803845039 0.37190923 12.5516.7219492 18.4627952 0.0521534012 0.41606511 12.619.7908763 21.0799858 0.0312408032 0.44952383 12.65

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21.8805802 22.6545722 0.018674916 0.47063971 12.722.8884697 23.1094403 0.015071127 0.47837761 12.7522.7654402 22.4228185 0.02060187 0.47236366 12.821.5182652 20.6293186 0.0349883837 0.45290335 12.8519.2092274 17.8181918 0.0575147559 0.42096596 12.915.9530091 14.1288877 0.0870635473 0.37813637 12.9511.910996 9.74413627 0.1221712211 0.32653633 13

7.28327906 4.88089824 0.161100631 0.26871942 13.052.29874953 -0.22037074 0.2019269743 0.20754464 13.1-2.79622233 -5.30760876 0.2426329608 0.14603504 13.15-7.74996431 -10.1296023 0.2812084885 0.08722834 13.2-12.3179339 -14.4483916 0.3157499043 0.03402692 13.25-16.2747899 -18.0510131 0.3445539457 -0.01094545 13.3-19.4255089 -20.7600014 0.3662017293 -0.04547298 13.35-21.6149971 -22.4421304 0.3796286433 -0.06785706 13.4-22.7357246 -23.0149658 0.3841767027 -0.07699978 13.45-22.7330064 -22.4509038 0.3796267916 -0.07245784 13.5-21.6076701 -20.7785012 0.3662092124 -0.05446396 13.55-19.4159807 -18.08103 0.3445920361 -0.02391505 13.6-16.2668284 -14.4923315 0.3158478402 0.01767257 13.65-12.3163179 -10.1901758 0.2814004926 0.06823836 13.7-7.7600312 -5.38745579 0.2429546203 0.1252797 13.75

-2.82334442 -0.32165225 0.2024112563 0.18597574 13.82.24972019 4.75690883 0.1617738406 0.24732687 13.857.20855625 9.59742595 0.123049221 0.306303 13.911.808353 13.9610077 0.0881485477 0.3599932 13.95

15.8221811 17.6324624 0.0587929539 0.40574934 1419.0521882 20.4309106 0.0364286792 0.44131676 14.0521.3393511 22.2186958 0.0221558192 0.46494533 14.122.5713052 22.9081559 0.0166742141 0.47547557 14.1522.6878644 22.4659213 0.0202491351 0.47239555 14.221.6839611 20.9145293 0.0326984529 0.45586577 14.2519.6098625 18.3312765 0.0534019082 0.42671083 14.316.5686538 14.8443681 0.0813320115 0.38637826 14.3512.7111153 10.626555 0.1151050358 0.3368667 14.48.22824709 5.88657604 0.153049568 0.28062684 14.453.34181165 0.85882681 0.1932892256 0.2204401 14.5-1.70663601 -4.20823065 0.2338354447 0.15928113 14.55-6.66768935 -9.06434651 0.2726857497 0.10017071 14.6-11.2964115 -13.4698423 0.3079226501 0.04602663 14.65-15.3644283 -17.2074404 0.3378082787 -0.0004804 14.7-18.6711958 -20.0929799 0.3608701019 -0.03705844 14.75-21.0538861 -21.9844906 0.3759734736 -0.06190729 14.8-22.3954058 -22.7891779 0.3823774586 -0.07380709 14.85-22.6301494 -22.4679749 0.3797711792 -0.07217818 14.9-21.7472076 -21.0374383 0.3682889021 -0.05710934 14.95-19.7908726 -18.5688968 0.3485031323 -0.02935303 15-16.858416 -15.1848945 0.3213960717 0.00971225 15.05

-13.0952533 -11.0531079 0.2883108618 0.05815036 15.1-8.68773273 -6.37803785 0.250885033 0.11356352 15.15-3.85390775 -1.39088777 0.2109694569 0.17321094 15.2

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1.16725065 3.66187026 0.1705368135 0.23414448 15.256.12766794 8.53067634 0.1315840997 0.29335425 15.310.7824222 12.9752087 0.0960339966 0.34791739 15.3514.9018373 16.7762456 0.0656399682 0.39514228 15.418.2828154 19.7464783 0.0418997749 0.43270134 15.4520.7588498 21.7397413 0.0259816675 0.45874578 15.522.2082258 22.6582044 0.0186668992 0.47199654 15.5522.5600025 22.4571749 0.0203113841 0.47180718 15.621.7974857 21.1472709 0.0308283832 0.45819533 15.6519.9590184 18.7938637 0.0496930616 0.43184147 15.717.1360534 15.5138152 0.0759686763 0.39405489 15.7513.4686041 11.4696752 0.1083530897 0.34670865 15.89.13830073 6.86162636 0.1452432979 0.29214675 15.854.35939501 1.91757518 0.184814778 0.23306806 15.9-0.63184042 -3.11811986 0.225111724 0.17239284 15.95-5.58879325 -7.9967254 0.2641437116 0.11311839 16-10.2666964 -12.4774191 0.2999840124 0.05817098 16.05-14.434714 -16.3391774 0.3308647012 0.01026139 16.1

-17.8873325 -19.3916777 0.3552638638 -0.0282488 16.15-20.4544932 -21.4846784 0.3719806013 -0.05546365 16.2-22.0099691 -22.5154136 0.3801941356 -0.07004619 16.25-22.4775708 -22.4336381 0.3795041075 -0.07128409 16.3-21.8348783 -21.2440769 0.3699500875 -0.05912446 16.35-20.1143147 -19.0061577 0.3520093509 -0.03417604 16.4-17.4015116 -15.8310422 0.3265730378 0.00232126 16.45-13.8310472 -11.8761049 0.294901891 0.04855796 16.5-9.57977043 -7.33713306 0.258563769 0.10224476 16.55-4.8580415 -2.43863438 0.2193560309 0.16072586 16.60.10067698 2.57726763 0.179216636 0.22111033 16.655.05136329 7.46280116 0.1401283556 0.28041504 16.79.74954357 11.9767853 0.1040208301 0.33571203 16.7513.9633643 15.8965361 0.0726753112 0.38427305 16.817.4850341 19.028852 0.0476367896 0.42370417 16.8520.1410703 21.2195365 0.0301378452 0.45206382 16.921.8008444 22.3609885 0.021037972 0.46795835 16.9522.3830072 22.3974876 0.0207813633 0.47061052 1721.8594748 21.3279131 0.0293752333 0.45989756 17.0520.2567832 19.2057665 0.0463897302 0.43635673 17.117.6547438 16.1364948 0.0709794331 0.40115847 17.1514.1824694 12.272252 0.101925358 0.35604816 17.210.0119679 7.80435584 0.1376953851 0.3032596 17.255.34962124 2.95381614 0.1765201126 0.24540437 17.30.42597276 -2.03959761 0.2164803807 0.18534265 17.35-4.51567264 -6.92920842 0.2556021383 0.12604183 17.4-9.2312715 -11.4736176 0.2919539605 0.07042986 17.45

-13.4880938 -15.4486226 0.3237424017 0.02125075 17.5-17.0762088 -18.6582778 0.3494004755 -0.01907086 17.55-19.8188381 -20.9445538 0.3676648979 -0.04854925 17.6-21.5810647 -22.1951177 0.3776382835 -0.06573538 17.65

-22.27647 -22.3488525 0.3788332355 -0.06978815 17.7-21.8713714 -21.3988432 0.371196162 -0.06051552 17.75

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-20.3864529 -19.3926852 0.3551096539 -0.03838362 17.8-17.8957106 -16.4301001 0.3313733233 -0.00449333 17.85-14.5227649 -12.6579788 0.3011640583 0.0394745 17.9-10.4347261 -8.26309907 0.2659776723 0.09134239 17.95-5.83391419 -3.46287672 0.2275548399 0.14854409 18-0.94784677 1.50538958 0.1877949861 0.20825153 18.053.98201237 6.39624916 0.1486623895 0.26751463 18.18.7121859 10.9682254 0.1120891407 0.32340709 18.15

13.0092078 14.995738 0.0798797499 0.37317086 18.216.6611463 18.2802332 0.0536221141 0.41435226 18.2519.4880567 20.6599719 0.0346092358 0.44492293 18.321.3508472 22.0179944 0.0237755532 0.46337974 18.3522.1581232 22.2878682 0.0216510158 0.46881873 18.421.8706705 21.4569378 0.0283351641 0.46097934 18.45

20.50336 19.5669164 0.0434924795 0.44025689 18.518.1243798 16.711792 0.0663692214 0.40768267 18.5514.851835 13.0331545 0.0958309078 0.3648726 18.6

10.8478846 8.71317364 0.1304185736 0.31394708 18.656.3107064 3.96557796 0.1684210188 0.25742597 18.71.464688 -0.97491895 0.2079594683 0.19810388 18.75

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-20.6075475 -19.7284695 0.3578039447 -0.04197677 19.2-18.3407267 -16.9815123 0.3357924631 -0.01072591 19.25-15.1695883 -13.3976557 0.307088753 0.0309119 19.3-11.2512902 -9.15439704 0.2731144147 0.08087406 19.35-6.77979001 -4.46168741 0.2355504495 0.13668816 19.4-1.9762445 0.44845661 0.1962540871 0.19559433 19.452.92192883 5.3334241 0.1571669616 0.25468171 19.57.67278601 9.95199717 0.1202191762 0.31103228 19.5512.0418073 14.0762621 0.0872339959 0.36186498 19.615.8134737 17.5028562 0.059837876 0.40467297 19.6518.8018985 20.0629951 0.0393802647 0.43734721 19.720.8599883 21.6307869 0.0268671337 0.45828041 19.7521.8866829 22.1294219 0.0229115136 0.46644607 19.821.8319179 21.5349361 0.0277034624 0.46144884 19.8520.6990652 19.8773615 0.0410009412 0.44354362 19.918.5447337 17.2392099 0.0621420338 0.41362258 19.9515.4759406 13.7513659 0.0900778924 0.37317072 2011.6447961 9.58659336 0.1234247721 0.32419217