Test dei solutori di sistemi di equazioni differenziali ordinarie (ODE)
Si consideri il seguente sistema di quattro equazioni differenziali:
`y[1] = y[3]
`y[2] = y[4]
`y[3] = −y[1] / ( y[1]2 +
y[2]2
)3/2
`y[4] = −y[2] / ( y[1]2 +
y[2]2
)3/2
Le condizioni iniziali al tempo t = 0, sono:
y[1] = 1 − e
y[2] = 0
y[3] = 0
y[4] = ((1 + e) / (1 − e))1/2
Il semiasse maggiore a vale 1 e
il periodo T vale 2·π
Eccentricità dell'orbita: e= 0.8
Passo di integrazione numerica: 0.002617993877991494
Vengono riportati i valori di x ed y ottenuti numericamente.
n x y vx vy t
0 0.200000000 0.000000000 -0.00000 3.00000 0.000000000
0 0.199999805 -0.000372785
1 0.198754401 0.029937818 -0.24824 2.98141 0.010000000
1 0.198785057 0.029567315
2 0.195069180 0.059509978 -0.48632 2.92747 0.020000000
2 0.195129432 0.059146153
3 0.189090406 0.088385864 -0.70575 2.84320 0.030000000
3 0.189177940 0.088032490
4 0.181035755 0.116296108 -0.90080 2.73559 0.040000000
4 0.181147547 0.115956091
5 0.171164929 0.143045186 -1.06877 2.61220 0.050000000
5 0.171297614 0.142720492
6 0.159751240 0.168510892 -1.20953 2.47999 0.060000000
6 0.159901436 0.168202623
7 0.147059558 0.192634758 -1.32476 2.34467 0.070000000
7 0.147224091 0.192343304
8 0.133332243 0.215408293 -1.41715 2.21052 0.080000000
8 0.133508274 0.215133512
9 0.118782221 0.236858868 -1.48982 2.08046 0.090000000
9 0.118967295 0.236600252
10 0.103591233 0.257037426 -1.54585 1.95634 0.100000000
10 0.103783281 0.256794239
11 0.087911290 0.276008831 -1.58806 1.83914 0.110000000
11 0.088108593 0.275780213
12 0.071867749 0.293844834 -1.61895 1.72929 0.120000000
12 0.072068898 0.293629873
13 0.055562984 0.310619241 -1.64063 1.62681 0.130000000
13 0.055766833 0.310417021
14 0.039080052 0.326404779 -1.65485 1.53147 0.140000000
14 0.039285674 0.326214412
15 0.022486031 0.341271186 -1.66306 1.44291 0.150000000
15 0.022692677 0.341091829
16 0.005834921 0.355284152 -1.66644 1.36070 0.160000000
16 0.006041991 0.355115016
17 -0.010829909 0.368504822 -1.66595 1.28437 0.170000000
17 -0.010622897 0.368345175
18 -0.027473701 0.380989662 -1.66235 1.21346 0.180000000
18 -0.027267134 0.380838831
19 -0.044068665 0.392790547 -1.65628 1.14751 0.190000000
19 -0.043862852 0.392647916
20 -0.060592639 0.403954987 -1.64823 1.08610 0.200000000
20 -0.060387824 0.403819991
21 -0.077028013 0.414526407 -1.63862 1.02884 0.210000000
21 -0.076824392 0.414398530
22 -0.093360862 0.424544469 -1.62777 0.97537 0.220000000
22 -0.093158587 0.424423240
23 -0.109580246 0.434045392 -1.61596 0.92536 0.230000000
23 -0.109379437 0.433930381
24 -0.125677648 0.443062266 -1.60341 0.87852 0.240000000
24 -0.125478399 0.442953079
25 -0.141646521 0.451625343 -1.59028 0.83456 0.250000000
25 -0.141448902 0.451521621
26 -0.157481919 0.459762314 -1.57674 0.79326 0.260000000
26 -0.157285984 0.459663728
27 -0.173180204 0.467498556 -1.56288 0.75438 0.270000000
27 -0.172985991 0.467404803
28 -0.188738799 0.474857356 -1.54881 0.71774 0.280000000
28 -0.188546334 0.474768158
29 -0.204155997 0.481860118 -1.53461 0.68314 0.290000000
29 -0.203965296 0.481775221
30 -0.219430798 0.488526544 -1.52034 0.65044 0.300000000
30 -0.219241870 0.488445714
31 -0.234562777 0.494874801 -1.50605 0.61949 0.310000000
31 -0.234375624 0.494797820
32 -0.249551981 0.500921663 -1.49179 0.59014 0.320000000
32 -0.249366601 0.500848330
33 -0.264398837 0.506682644 -1.47759 0.56229 0.330000000
33 -0.264215221 0.506612774
34 -0.279104079 0.512172118 -1.46347 0.53582 0.340000000
34 -0.278922217 0.512105539
35 -0.293668691 0.517403419 -1.44947 0.51064 0.350000000
35 -0.293488570 0.517339971
36 -0.308093857 0.522388940 -1.43559 0.48665 0.360000000
36 -0.307915461 0.522328475
37 -0.322380917 0.527140213 -1.42185 0.46378 0.370000000
37 -0.322204229 0.527082593
38 -0.336531336 0.531667986 -1.40826 0.44194 0.380000000
38 -0.336356337 0.531613081
39 -0.350546677 0.535982290 -1.39483 0.42107 0.390000000
39 -0.350373346 0.535929980
40 -0.364428573 0.540092503 -1.38157 0.40111 0.400000000
40 -0.364256890 0.540042675
41 -0.378178711 0.544007397 -1.36848 0.38200 0.410000000
41 -0.378008655 0.543959946
42 -0.391798819 0.547735197 -1.35557 0.36369 0.420000000
42 -0.391630368 0.547690024
43 -0.405290645 0.551283619 -1.34283 0.34612 0.430000000
43 -0.405123777 0.551240631
44 -0.418655952 0.554659912 -1.33026 0.32925 0.440000000
44 -0.418490646 0.554619021
45 -0.431896508 0.557870894 -1.31788 0.31305 0.450000000
45 -0.431732741 0.557832018
46 -0.445014076 0.560922987 -1.30567 0.29747 0.460000000
46 -0.444851828 0.560886049
47 -0.458010413 0.563822245 -1.29363 0.28248 0.470000000
47 -0.457849660 0.563787171
48 -0.470887257 0.566574381 -1.28177 0.26804 0.480000000
48 -0.470727979 0.566541103
49 -0.483646332 0.569184793 -1.27008 0.25413 0.490000000
49 -0.483488507 0.569153245
50 -0.496289337 0.571658587 -1.25855 0.24071 0.500000000
50 -0.496132945 0.571628708
51 -0.508817949 0.574000596 -1.24720 0.22777 0.510000000
51 -0.508662968 0.573972328
52 -0.521233816 0.576215402 -1.23600 0.21527 0.520000000
52 -0.521080227 0.576188688
53 -0.533538562 0.578307348 -1.22497 0.20319 0.530000000
53 -0.533386343 0.578282137
54 -0.545733778 0.580280560 -1.21410 0.19152 0.540000000
54 -0.545582911 0.580256800
55 -0.557821026 0.582138956 -1.20338 0.18022 0.550000000
55 -0.557671492 0.582116602
56 -0.569801840 0.583886265 -1.19281 0.16930 0.560000000
56 -0.569653620 0.583865270
57 -0.581677719 0.585526035 -1.18239 0.15871 0.570000000
57 -0.581530794 0.585506357
58 -0.593450134 0.587061645 -1.17212 0.14846 0.580000000
58 -0.593304486 0.587043242
59 -0.605120523 0.588496318 -1.16198 0.13852 0.590000000
59 -0.604976135 0.588479152
60 -0.616690294 0.589833128 -1.15199 0.12889 0.600000000
60 -0.616547148 0.589817161
61 -0.628160823 0.591075010 -1.14214 0.11954 0.610000000
61 -0.628018902 0.591060206
62 -0.639533457 0.592224768 -1.13241 0.11046 0.620000000
62 -0.639392745 0.592211094
63 -0.650809512 0.593285084 -1.12282 0.10165 0.630000000
63 -0.650669993 0.593272507
64 -0.661990275 0.594258524 -1.11335 0.09308 0.640000000
64 -0.661851932 0.594247013
65 -0.673077003 0.595147544 -1.10401 0.08476 0.650000000
65 -0.672939821 0.595137068
66 -0.684070924 0.595954496 -1.09479 0.07667 0.660000000
66 -0.683934889 0.595945028
67 -0.694973240 0.596681638 -1.08569 0.06880 0.670000000
67 -0.694838337 0.596673149
68 -0.705785124 0.597331131 -1.07671 0.06114 0.680000000
68 -0.705651337 0.597323596
69 -0.716507722 0.597905055 -1.06783 0.05368 0.690000000
69 -0.716375039 0.597898447
70 -0.727142154 0.598405401 -1.05907 0.04642 0.700000000
70 -0.727010561 0.598399697
71 -0.737689516 0.598834088 -1.05042 0.03935 0.710000000
71 -0.737558998 0.598829265
72 -0.748150876 0.599192958 -1.04187 0.03246 0.720000000
72 -0.748021421 0.599188992
73 -0.758527280 0.599483782 -1.03343 0.02574 0.730000000
73 -0.758398876 0.599480653
74 -0.768819750 0.599708267 -1.02508 0.01919 0.740000000
74 -0.768692383 0.599705953
75 -0.779029283 0.599868054 -1.01684 0.01280 0.750000000
75 -0.778902941 0.599866536
76 -0.789156855 0.599964727 -1.00869 0.00656 0.760000000
76 -0.789031526 0.599963985
77 -0.799203419 0.599999810 -1.00064 0.00048 0.770000000
77 -0.799079092 0.599999826
78 -0.809169909 0.599974773 -0.99268 -0.00546 0.780000000
78 -0.809046572 0.599975529
79 -0.819057235 0.599891037 -0.98480 -0.01126 0.790000000
79 -0.818934877 0.599892515
80 -0.828866288 0.599749969 -0.97702 -0.01693 0.800000000
80 -0.828744898 0.599752153
81 -0.838597940 0.599552893 -0.96932 -0.02247 0.810000000
81 -0.838477507 0.599555766
82 -0.848253043 0.599301086 -0.96171 -0.02788 0.820000000
82 -0.848133557 0.599304633
83 -0.857832430 0.598995783 -0.95418 -0.03317 0.830000000
83 -0.857713881 0.598999988
84 -0.867336917 0.598638176 -0.94673 -0.03834 0.840000000
84 -0.867219294 0.598643026
85 -0.876767301 0.598229422 -0.93936 -0.04340 0.850000000
85 -0.876650594 0.598234902
86 -0.886124362 0.597770637 -0.93207 -0.04834 0.860000000
86 -0.886008562 0.597776733
87 -0.895408864 0.597262902 -0.92485 -0.05319 0.870000000
87 -0.895293962 0.597269601
88 -0.904621554 0.596707264 -0.91770 -0.05792 0.880000000
88 -0.904507541 0.596714554
89 -0.913763163 0.596104739 -0.91063 -0.06256 0.890000000
89 -0.913650030 0.596112607
90 -0.922834408 0.595456308 -0.90363 -0.06711 0.900000000
90 -0.922722145 0.595464743
91 -0.931835989 0.594762926 -0.89670 -0.07155 0.910000000
91 -0.931724588 0.594771915
92 -0.940768592 0.594025516 -0.88983 -0.07591 0.920000000
92 -0.940658046 0.594035047
93 -0.949632890 0.593244974 -0.88304 -0.08018 0.930000000
93 -0.949523189 0.593255038
94 -0.958429541 0.592422171 -0.87630 -0.08436 0.940000000
94 -0.958320677 0.592432756
95 -0.967159188 0.591557952 -0.86964 -0.08847 0.950000000
95 -0.967051155 0.591569048
96 -0.975822465 0.590653135 -0.86303 -0.09248 0.960000000
96 -0.975715254 0.590664732
97 -0.984419990 0.589708518 -0.85649 -0.09643 0.970000000
97 -0.984313592 0.589720607
98 -0.992952368 0.588724875 -0.85000 -0.10029 0.980000000
98 -0.992846778 0.588737445
99 -1.001420195 0.587702957 -0.84357 -0.10408 0.990000000
99 -1.001315404 0.587716000
100 -1.009824052 0.586643497 -0.83721 -0.10780 1.000000000
100 -1.009720053 0.586657003
101 -1.018164510 0.585547205 -0.83089 -0.11145 1.010000000
101 -1.018061297 0.585561166
102 -1.026442129 0.584414773 -0.82464 -0.11503 1.020000000
102 -1.026339694 0.584429180
103 -1.034657458 0.583246874 -0.81844 -0.11854 1.030000000
103 -1.034555795 0.583261720
104 -1.042811034 0.582044164 -0.81229 -0.12199 1.040000000
104 -1.042710136 0.582059440
105 -1.050903386 0.580807280 -0.80619 -0.12538 1.050000000
105 -1.050803246 0.580822979
106 -1.058935030 0.579536844 -0.80015 -0.12870 1.060000000
106 -1.058835643 0.579552958
107 -1.066906474 0.578233462 -0.79415 -0.13197 1.070000000
107 -1.066807833 0.578249982
108 -1.074818217 0.576897721 -0.78821 -0.13517 1.080000000
108 -1.074720316 0.576914642
109 -1.082670748 0.575530199 -0.78231 -0.13832 1.090000000
109 -1.082573580 0.575547512
110 -1.090464544 0.574131453 -0.77646 -0.14142 1.100000000
110 -1.090368105 0.574149153
111 -1.098200078 0.572702031 -0.77066 -0.14446 1.110000000
111 -1.098104361 0.572720110
112 -1.105877811 0.571242466 -0.76490 -0.14745 1.120000000
112 -1.105782811 0.571260918
113 -1.113498196 0.569753277 -0.75919 -0.15038 1.130000000
113 -1.113403907 0.569772096
114 -1.121061679 0.568234972 -0.75352 -0.15327 1.140000000
114 -1.120968095 0.568254150
115 -1.128568696 0.566688045 -0.74789 -0.15611 1.150000000
115 -1.128475812 0.566707578
116 -1.136019676 0.565112980 -0.74231 -0.15890 1.160000000
116 -1.135927487 0.565132861
117 -1.143415040 0.563510248 -0.73677 -0.16164 1.170000000
117 -1.143323541 0.563530471
118 -1.150755203 0.561880311 -0.73127 -0.16434 1.180000000
118 -1.150664389 0.561900871
119 -1.158040571 0.560223618 -0.72581 -0.16699 1.190000000
119 -1.157950436 0.560244509
120 -1.165271543 0.558540609 -0.72039 -0.16960 1.200000000
120 -1.165182083 0.558561826
121 -1.172448511 0.556831715 -0.71501 -0.17217 1.210000000
121 -1.172359720 0.556853253
122 -1.179571860 0.555097355 -0.70967 -0.17470 1.220000000
122 -1.179483735 0.555119208
123 -1.186641969 0.553337940 -0.70436 -0.17718 1.230000000
123 -1.186554504 0.553360103
124 -1.193659210 0.551553872 -0.69909 -0.17963 1.240000000
124 -1.193572401 0.551576341
125 -1.200623949 0.549745544 -0.69386 -0.18203 1.250000000
125 -1.200537792 0.549768313
126 -1.207536544 0.547913339 -0.68866 -0.18440 1.260000000
126 -1.207451034 0.547936405
127 -1.214397350 0.546057635 -0.68350 -0.18673 1.270000000
127 -1.214312483 0.546080992
128 -1.221206713 0.544178799 -0.67838 -0.18903 1.280000000
128 -1.221122484 0.544202443
129 -1.227964975 0.542277192 -0.67328 -0.19129 1.290000000
129 -1.227881380 0.542301118
130 -1.234672470 0.540353166 -0.66822 -0.19351 1.300000000
130 -1.234589506 0.540377370
131 -1.241329530 0.538407066 -0.66320 -0.19570 1.310000000
131 -1.241247192 0.538431543
132 -1.247936479 0.536439230 -0.65820 -0.19786 1.320000000
132 -1.247854762 0.536463978
133 -1.254493635 0.534449991 -0.65324 -0.19998 1.330000000
133 -1.254412537 0.534475004
134 -1.261001313 0.532439672 -0.64830 -0.20208 1.340000000
134 -1.260920829 0.532464946
135 -1.267459821 0.530408591 -0.64340 -0.20414 1.350000000
135 -1.267379947 0.530434123
136 -1.273869463 0.528357061 -0.63853 -0.20617 1.360000000
136 -1.273790196 0.528382846
137 -1.280230537 0.526285386 -0.63369 -0.20816 1.370000000
137 -1.280151873 0.526311421
138 -1.286543338 0.524193866 -0.62888 -0.21013 1.380000000
138 -1.286465274 0.524220148
139 -1.292808154 0.522082795 -0.62409 -0.21208 1.390000000
139 -1.292730686 0.522109319
140 -1.299025270 0.519952460 -0.61934 -0.21399 1.400000000
140 -1.298948395 0.519979224
141 -1.305194966 0.517803145 -0.61461 -0.21587 1.410000000
141 -1.305118680 0.517830144
142 -1.311317518 0.515635125 -0.60991 -0.21773 1.420000000
142 -1.311241817 0.515662357
143 -1.317393196 0.513448674 -0.60523 -0.21956 1.430000000
143 -1.317318077 0.513476135
144 -1.323422267 0.511244058 -0.60059 -0.22136 1.440000000
144 -1.323347727 0.511271744
145 -1.329404993 0.509021539 -0.59596 -0.22314 1.450000000
145 -1.329331029 0.509049447
146 -1.335341635 0.506781373 -0.59137 -0.22489 1.460000000
146 -1.335268243 0.506809501
147 -1.341232444 0.504523814 -0.58680 -0.22662 1.470000000
147 -1.341159623 0.504552158
148 -1.347077674 0.502249108 -0.58225 -0.22832 1.480000000
148 -1.347005418 0.502277665
149 -1.352877569 0.499957500 -0.57773 -0.23000 1.490000000
149 -1.352805877 0.499986267
150 -1.358632372 0.497649228 -0.57323 -0.23165 1.500000000
150 -1.358561241 0.497678202
151 -1.364342324 0.495324527 -0.56876 -0.23328 1.510000000
151 -1.364271750 0.495353705
152 -1.370007659 0.492983627 -0.56431 -0.23489 1.520000000
152 -1.369937639 0.493013007
153 -1.375628608 0.490626756 -0.55988 -0.23648 1.530000000
153 -1.375559140 0.490656334
154 -1.381205401 0.488254136 -0.55548 -0.23804 1.540000000
154 -1.381136482 0.488283910
155 -1.386738262 0.485865986 -0.55110 -0.23958 1.550000000
155 -1.386669890 0.485895953
156 -1.392227413 0.483462521 -0.54674 -0.24111 1.560000000
156 -1.392159584 0.483492678
157 -1.397673071 0.481043952 -0.54240 -0.24261 1.570000000
157 -1.397605783 0.481074298
158 -1.403075452 0.478610489 -0.53808 -0.24408 1.580000000
158 -1.403008701 0.478641020
159 -1.408434767 0.476162334 -0.53379 -0.24554 1.590000000
159 -1.408368552 0.476193048
160 -1.413751224 0.473699690 -0.52951 -0.24698 1.600000000
160 -1.413685542 0.473730585
161 -1.419025029 0.471222755 -0.52525 -0.24840 1.610000000
161 -1.418959877 0.471253827
162 -1.424256383 0.468731723 -0.52102 -0.24980 1.620000000
162 -1.424191760 0.468762971
163 -1.429445488 0.466226786 -0.51680 -0.25118 1.630000000
163 -1.429381390 0.466258207
164 -1.434592538 0.463708132 -0.51261 -0.25254 1.640000000
164 -1.434528963 0.463739724
165 -1.439697727 0.461175948 -0.50843 -0.25389 1.650000000
165 -1.439634672 0.461207708
166 -1.444761245 0.458630415 -0.50427 -0.25521 1.660000000
166 -1.444698709 0.458662341
167 -1.449783281 0.456071713 -0.50014 -0.25652 1.670000000
167 -1.449721262 0.456103803
168 -1.454764020 0.453500020 -0.49602 -0.25781 1.680000000
168 -1.454702514 0.453532272
169 -1.459703644 0.450915510 -0.49191 -0.25909 1.690000000
169 -1.459642649 0.450947922
170 -1.464602332 0.448318354 -0.48783 -0.26034 1.700000000
170 -1.464541847 0.448350924
171 -1.469460262 0.445708722 -0.48376 -0.26158 1.710000000
171 -1.469400284 0.445741447
172 -1.474277608 0.443086780 -0.47971 -0.26280 1.720000000
172 -1.474218135 0.443119659
173 -1.479054542 0.440452693 -0.47568 -0.26401 1.730000000
173 -1.478995572 0.440485722
174 -1.483791234 0.437806621 -0.47166 -0.26520 1.740000000
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