This report gives the result of solving the 1,940 Kamke differential equations in Mathematica 10.0.2 and Maple 18.02 on windows 7, 64 bit OS. The PC used is an Intel i7-3930k running at 3.20 GHz with 16 GB memory. Sage 6.2 is currently work in progress and is being added. Currently only the first 70 Kamke differential equations are implemented in Sage.
The command AboluteTiming[] was used in Mathematica to obtain the CPU time. In Sage
the command cputime() was used. In Maple the following command was used for this
purpose
cpu_time := Usage(assign ('result_of_solve',dsolve(ode,y(x))),output='cputime');
When Mathematica returned DifferentialRoot as a solution to an ODE this was counted
as not solved. Similarly, when Maple returned DESol this was also counted as not
solved.
Table 1 below summarizes the performance of each CAS system
A number of problems were not solved due to the program hanging. When a program took more than 20 minutes of real time or more (sometimes, the wait was over one hour before terminating the process), only then it was considered as hanged process. Also, when it causes the PC to run of memory while running forcing the termination of the current process, this was counted as hanged process as well. Table 2 shows the percentage that hanged out of all the unsolved problems.
Table 3 summarizes the Kamke equations used
| book chapter | kamke book numbering of equations | Numbering in this document |
| Chapter 1, linear first order | 1.1—1.576 | 1—576 |
| Additional non-linear first order | N/A | 577—1000 |
| Chapter 2, linear second order | 2.1—2.448 | 1001—1448 |
| Chapter 3, linear third order | 3.1—3.85 | 1449—1533 |
| Chapter 4, linear fourth order | 4.1—4.44 | 1534—1577 |
| Chapter 5, linear fifth and higher order | 5.1—5.13 | 1578—1590 |
| Chapter 6, non-linear second order | 6.1—6.246 | 1591—1836 |
| Chapter 7, non-linear third and higher order | 7.1—7.19 | 1837—1855 |
| Chapter 8, system of ode, first order | 8.1—8.57 | 1856—1912 |
| Chapter 9, system of ode, higher order | 9.1—9.28 | 1913–1940 |
The following summarizes which equations are solved by each system
14, 23, 24, 31, 35, 36, 37, 38, 40, 41, 42, 43, 45, 46, 47, 48, 49, 50, 52, 55, 56, 58, 59, 62, 63, 64, 65, 67, 68, 69
16, 22, 38, 47, 48, 49, 50, 55, 56, 74, 79, 80, 81, 82, 83, 86, 87, 110, 121, 127, 188, 202, 203, 205, 206, 219, 234, 237, 250, 253, 265, 266, 269, 331, 340, 365, 367, 368, 370, 383, 385, 394, 395, 400, 402, 404, 413, 414, 416, 428, 429, 430, 451, 452, 460, 461, 465, 467, 468, 479, 480, 482, 485, 487, 489, 494, 503, 504, 506, 509, 510, 513, 515, 523, 524, 527, 528, 530, 531, 533, 534, 535, 537, 538, 541, 542, 543, 544, 546, 550, 555, 561, 566, 567, 570, 572, 575, 576, 620, 638, 639, 640, 672, 701, 703, 706, 714, 730, 733, 735, 743, 745, 746, 747, 752, 759, 766, 769, 776, 782, 783, 784, 785, 786, 788, 789, 807, 835, 837, 854, 855, 862, 885, 889, 892, 894, 909, 913, 915, 916, 917, 918, 919, 922, 923, 925, 926, 929, 932, 942, 953, 961, 996, 1000, 1008, 1026, 1036, 1049, 1068, 1073, 1090, 1156, 1212, 1216, 1232, 1233, 1236, 1263, 1278, 1341, 1413, 1418, 1427, 1439, 1441, 1442, 1443, 1444, 1445, 1450, 1457, 1458, 1459, 1460, 1461, 1462, 1463, 1471, 1472, 1473, 1474, 1476, 1482, 1484, 1489, 1500, 1505, 1507, 1510, 1515, 1516, 1520, 1526, 1527, 1529, 1530, 1531, 1540, 1541, 1542, 1543, 1544, 1547, 1552, 1572, 1573, 1574, 1575, 1576, 1581, 1586, 1590, 1593, 1595, 1596, 1598, 1599, 1601, 1603, 1605, 1606, 1608, 1609, 1610, 1611, 1612, 1613, 1614, 1615, 1616, 1617, 1618, 1619, 1620, 1621, 1622, 1623, 1624, 1625, 1626, 1627, 1628, 1629, 1631, 1633, 1634, 1635, 1636, 1637, 1638, 1639, 1640, 1642, 1643, 1644, 1645, 1648, 1649, 1652, 1656, 1658, 1659, 1660, 1662, 1663, 1664, 1665, 1666, 1667, 1672, 1673, 1675, 1677, 1678, 1680, 1681, 1682, 1684, 1685, 1686, 1690, 1691, 1692, 1693, 1695, 1696, 1702, 1704, 1705, 1706, 1708, 1709, 1710, 1713, 1719, 1720, 1721, 1729, 1732, 1734, 1735, 1737, 1738, 1739, 1742, 1746, 1751, 1755, 1757, 1761, 1762, 1776, 1777, 1779, 1780, 1788, 1789, 1797, 1798, 1801, 1802, 1806, 1807, 1809, 1811, 1813, 1815, 1816, 1817, 1818, 1819, 1820, 1821, 1823, 1825, 1827, 1831, 1832, 1833, 1834, 1836, 1837, 1838, 1839, 1840, 1841, 1844, 1845, 1848, 1850, 1851, 1853, 1854, 1855, 1875, 1880, 1885, 1890, 1894, 1905, 1911, 1912, 1915, 1916, 1917, 1918, 1919, 1920, 1921, 1922, 1925, 1926, 1927, 1928, 1929, 1932, 1933, 1934, 1935, 1936, 1937, 1939, 1940
38, 47, 48, 49, 50, 55, 56, 74, 79, 82, 87, 110, 121, 202, 203, 205, 206, 219, 234, 237, 250, 253, 265, 269, 286, 331, 340, 367, 368, 370, 383, 395, 452, 460, 461, 480, 482, 484, 485, 503, 506, 507, 510, 531, 537, 543, 572, 575, 576, 733, 788, 789, 790, 835, 837, 885, 894, 920, 1015, 1019, 1026, 1028, 1031, 1038, 1072, 1073, 1075, 1076, 1077, 1081, 1158, 1212, 1216, 1236, 1278, 1457, 1458, 1459, 1460, 1461, 1462, 1463, 1473, 1474, 1476, 1484, 1489, 1510, 1515, 1531, 1540, 1541, 1542, 1543, 1581, 1582, 1586, 1593, 1595, 1596, 1598, 1599, 1606, 1608, 1609, 1616, 1619, 1623, 1625, 1628, 1634, 1642, 1643, 1645, 1649, 1675, 1685, 1698, 1702, 1704, 1705, 1706, 1721, 1729, 1732, 1734, 1735, 1737, 1738, 1739, 1751, 1757, 1761, 1788, 1789, 1797, 1801, 1802, 1807, 1835, 1851, 1854, 1855, 1880, 1890, 1905, 1921, 1922, 1927, 1928, 1940
286, 484, 507, 790, 920, 1015, 1019, 1028, 1031, 1038, 1072, 1075, 1076, 1077, 1081, 1158, 1582, 1698, 1835
16, 22, 80, 81, 83, 86, 127, 188, 266, 365, 385, 394, 400, 402, 404, 413, 414, 416, 428, 429, 430, 451, 465, 467, 468, 479, 487, 489, 494, 504, 509, 513, 515, 523, 524, 527, 528, 530, 533, 534, 535, 538, 541, 542, 544, 546, 550, 555, 561, 566, 567, 570, 620, 638, 639, 640, 672, 701, 703, 706, 714, 730, 735, 743, 745, 746, 747, 752, 759, 766, 769, 776, 782, 783, 784, 785, 786, 807, 854, 855, 862, 889, 892, 909, 913, 915, 916, 917, 918, 919, 922, 923, 925, 926, 929, 932, 942, 953, 961, 996, 1000, 1008, 1036, 1049, 1068, 1090, 1156, 1232, 1233, 1263, 1341, 1413, 1418, 1427, 1439, 1441, 1442, 1443, 1444, 1445, 1450, 1471, 1472, 1482, 1500, 1505, 1507, 1516, 1520, 1526, 1527, 1529, 1530, 1544, 1547, 1552, 1572, 1573, 1574, 1575, 1576, 1590, 1601, 1603, 1605, 1610, 1611, 1612, 1613, 1614, 1615, 1617, 1618, 1620, 1621, 1622, 1624, 1626, 1627, 1629, 1631, 1633, 1635, 1636, 1637, 1638, 1639, 1640, 1644, 1648, 1652, 1656, 1658, 1659, 1660, 1662, 1663, 1664, 1665, 1666, 1667, 1672, 1673, 1677, 1678, 1680, 1681, 1682, 1684, 1686, 1690, 1691, 1692, 1693, 1695, 1696, 1708, 1709, 1710, 1713, 1719, 1720, 1742, 1746, 1755, 1762, 1776, 1777, 1779, 1780, 1798, 1806, 1809, 1811, 1813, 1815, 1816, 1817, 1818, 1819, 1820, 1821, 1823, 1825, 1827, 1831, 1832, 1833, 1834, 1836, 1837, 1838, 1839, 1840, 1841, 1844, 1845, 1848, 1850, 1853, 1875, 1885, 1894, 1911, 1912, 1915, 1916, 1917, 1918, 1919, 1920, 1925, 1926, 1929, 1932, 1933, 1934, 1935, 1936, 1937, 1939
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 43, 44, 45, 46, 51, 52, 53, 54, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 75, 76, 77, 78, 84, 85, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 122, 123, 124, 125, 126, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 204, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 235, 236, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 251, 252, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 267, 268, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 332, 333, 334, 335, 336, 337, 338, 339, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 366, 369, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 384, 386, 387, 388, 389, 390, 391, 392, 393, 396, 397, 398, 399, 401, 403, 405, 406, 407, 408, 409, 410, 411, 412, 415, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 453, 454, 455, 456, 457, 458, 459, 462, 463, 464, 466, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 481, 483, 486, 488, 490, 491, 492, 493, 495, 496, 497, 498, 499, 500, 501, 502, 505, 508, 511, 512, 514, 516, 517, 518, 519, 520, 521, 522, 525, 526, 529, 532, 536, 539, 540, 545, 547, 548, 549, 551, 552, 553, 554, 556, 557, 558, 559, 560, 562, 563, 564, 565, 568, 569, 571, 573, 574, 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 591, 592, 593, 594, 595, 596, 597, 598, 599, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 621, 622, 623, 624, 625, 626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 641, 642, 643, 644, 645, 646, 647, 648, 649, 650, 651, 652, 653, 654, 655, 656, 657, 658, 659, 660, 661, 662, 663, 664, 665, 666, 667, 668, 669, 670, 671, 673, 674, 675, 676, 677, 678, 679, 680, 681, 682, 683, 684, 685, 686, 687, 688, 689, 690, 691, 692, 693, 694, 695, 696, 697, 698, 699, 700, 702, 704, 705, 707, 708, 709, 710, 711, 712, 713, 715, 716, 717, 718, 719, 720, 721, 722, 723, 724, 725, 726, 727, 728, 729, 731, 732, 734, 736, 737, 738, 739, 740, 741, 742, 744, 748, 749, 750, 751, 753, 754, 755, 756, 757, 758, 760, 761, 762, 763, 764, 765, 767, 768, 770, 771, 772, 773, 774, 775, 777, 778, 779, 780, 781, 787, 791, 792, 793, 794, 795, 796, 797, 798, 799, 800, 801, 802, 803, 804, 805, 806, 808, 809, 810, 811, 812, 813, 814, 815, 816, 817, 818, 819, 820, 821, 822, 823, 824, 825, 826, 827, 828, 829, 830, 831, 832, 833, 834, 836, 838, 839, 840, 841, 842, 843, 844, 845, 846, 847, 848, 849, 850, 851, 852, 853, 856, 857, 858, 859, 860, 861, 863, 864, 865, 866, 867, 868, 869, 870, 871, 872, 873, 874, 875, 876, 877, 878, 879, 880, 881, 882, 883, 884, 886, 887, 888, 890, 891, 893, 895, 896, 897, 898, 899, 900, 901, 902, 903, 904, 905, 906, 907, 908, 910, 911, 912, 914, 921, 924, 927, 928, 930, 931, 933, 934, 935, 936, 937, 938, 939, 940, 941, 943, 944, 945, 946, 947, 948, 949, 950, 951, 952, 954, 955, 956, 957, 958, 959, 960, 962, 963, 964, 965, 966, 967, 968, 969, 970, 971, 972, 973, 974, 975, 976, 977, 978, 979, 980, 981, 982, 983, 984, 985, 986, 987, 988, 989, 990, 991, 992, 993, 994, 995, 997, 998, 999, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1009, 1010, 1011, 1012, 1013, 1014, 1016, 1017, 1018, 1020, 1021, 1022, 1023, 1024, 1025, 1027, 1029, 1030, 1032, 1033, 1034, 1035, 1037, 1039, 1040, 1041, 1042, 1043, 1044, 1045, 1046, 1047, 1048, 1050, 1051, 1052, 1053, 1054, 1055, 1056, 1057, 1058, 1059, 1060, 1061, 1062, 1063, 1064, 1065, 1066, 1067, 1069, 1070, 1071, 1074, 1078, 1079, 1080, 1082, 1083, 1084, 1085, 1086, 1087, 1088, 1089, 1091, 1092, 1093, 1094, 1095, 1096, 1097, 1098, 1099, 1100, 1101, 1102, 1103, 1104, 1105, 1106, 1107, 1108, 1109, 1110, 1111, 1112, 1113, 1114, 1115, 1116, 1117, 1118, 1119, 1120, 1121, 1122, 1123, 1124, 1125, 1126, 1127, 1128, 1129, 1130, 1131, 1132, 1133, 1134, 1135, 1136, 1137, 1138, 1139, 1140, 1141, 1142, 1143, 1144, 1145, 1146, 1147, 1148, 1149, 1150, 1151, 1152, 1153, 1154, 1155, 1157, 1159, 1160, 1161, 1162, 1163, 1164, 1165, 1166, 1167, 1168, 1169, 1170, 1171, 1172, 1173, 1174, 1175, 1176, 1177, 1178, 1179, 1180, 1181, 1182, 1183, 1184, 1185, 1186, 1187, 1188, 1189, 1190, 1191, 1192, 1193, 1194, 1195, 1196, 1197, 1198, 1199, 1200, 1201, 1202, 1203, 1204, 1205, 1206, 1207, 1208, 1209, 1210, 1211, 1213, 1214, 1215, 1217, 1218, 1219, 1220, 1221, 1222, 1223, 1224, 1225, 1226, 1227, 1228, 1229, 1230, 1231, 1234, 1235, 1237, 1238, 1239, 1240, 1241, 1242, 1243, 1244, 1245, 1246, 1247, 1248, 1249, 1250, 1251, 1252, 1253, 1254, 1255, 1256, 1257, 1258, 1259, 1260, 1261, 1262, 1264, 1265, 1266, 1267, 1268, 1269, 1270, 1271, 1272, 1273, 1274, 1275, 1276, 1277, 1279, 1280, 1281, 1282, 1283, 1284, 1285, 1286, 1287, 1288, 1289, 1290, 1291, 1292, 1293, 1294, 1295, 1296, 1297, 1298, 1299, 1300, 1301, 1302, 1303, 1304, 1305, 1306, 1307, 1308, 1309, 1310, 1311, 1312, 1313, 1314, 1315, 1316, 1317, 1318, 1319, 1320, 1321, 1322, 1323, 1324, 1325, 1326, 1327, 1328, 1329, 1330, 1331, 1332, 1333, 1334, 1335, 1336, 1337, 1338, 1339, 1340, 1342, 1343, 1344, 1345, 1346, 1347, 1348, 1349, 1350, 1351, 1352, 1353, 1354, 1355, 1356, 1357, 1358, 1359, 1360, 1361, 1362, 1363, 1364, 1365, 1366, 1367, 1368, 1369, 1370, 1371, 1372, 1373, 1374, 1375, 1376, 1377, 1378, 1379, 1380, 1381, 1382, 1383, 1384, 1385, 1386, 1387, 1388, 1389, 1390, 1391, 1392, 1393, 1394, 1395, 1396, 1397, 1398, 1399, 1400, 1401, 1402, 1403, 1404, 1405, 1406, 1407, 1408, 1409, 1410, 1411, 1412, 1414, 1415, 1416, 1417, 1419, 1420, 1421, 1422, 1423, 1424, 1425, 1426, 1428, 1429, 1430, 1431, 1432, 1433, 1434, 1435, 1436, 1437, 1438, 1440, 1446, 1447, 1448, 1449, 1451, 1452, 1453, 1454, 1455, 1456, 1464, 1465, 1466, 1467, 1468, 1469, 1470, 1475, 1477, 1478, 1479, 1480, 1481, 1483, 1485, 1486, 1487, 1488, 1490, 1491, 1492, 1493, 1494, 1495, 1496, 1497, 1498, 1499, 1501, 1502, 1503, 1504, 1506, 1508, 1509, 1511, 1512, 1513, 1514, 1517, 1518, 1519, 1521, 1522, 1523, 1524, 1525, 1528, 1532, 1533, 1534, 1535, 1536, 1537, 1538, 1539, 1545, 1546, 1548, 1549, 1550, 1551, 1553, 1554, 1555, 1556, 1557, 1558, 1559, 1560, 1561, 1562, 1563, 1564, 1565, 1566, 1567, 1568, 1569, 1570, 1571, 1577, 1578, 1579, 1580, 1583, 1584, 1585, 1587, 1588, 1589, 1591, 1592, 1594, 1597, 1600, 1602, 1604, 1607, 1630, 1632, 1641, 1646, 1647, 1650, 1651, 1653, 1654, 1655, 1657, 1661, 1668, 1669, 1670, 1671, 1674, 1676, 1679, 1683, 1687, 1688, 1689, 1694, 1697, 1699, 1700, 1701, 1703, 1707, 1711, 1712, 1714, 1715, 1716, 1717, 1718, 1722, 1723, 1724, 1725, 1726, 1727, 1728, 1730, 1731, 1733, 1736, 1740, 1741, 1743, 1744, 1745, 1747, 1748, 1749, 1750, 1752, 1753, 1754, 1756, 1758, 1759, 1760, 1763, 1764, 1765, 1766, 1767, 1768, 1769, 1770, 1771, 1772, 1773, 1774, 1775, 1778, 1781, 1782, 1783, 1784, 1785, 1786, 1787, 1790, 1791, 1792, 1793, 1794, 1795, 1796, 1799, 1800, 1803, 1804, 1805, 1808, 1810, 1812, 1814, 1822, 1824, 1826, 1828, 1829, 1830, 1842, 1843, 1846, 1847, 1849, 1852, 1856, 1857, 1858, 1859, 1860, 1861, 1862, 1863, 1864, 1865, 1866, 1867, 1868, 1869, 1870, 1871, 1872, 1873, 1874, 1876, 1877, 1878, 1879, 1881, 1882, 1883, 1884, 1886, 1887, 1888, 1889, 1891, 1892, 1893, 1895, 1896, 1897, 1898, 1899, 1900, 1901, 1902, 1903, 1904, 1906, 1907, 1908, 1909, 1910, 1913, 1914, 1923, 1924, 1930, 1931, 1938
38, 47, 48, 49, 50, 55, 56, 74, 79, 82, 87, 110, 121, 202, 203, 205, 206, 219, 234, 237, 250, 253, 265, 269, 331, 340, 367, 368, 370, 383, 395, 452, 460, 461, 480, 482, 485, 503, 506, 510, 531, 537, 543, 572, 575, 576, 733, 788, 789, 835, 837, 885, 894, 1026, 1073, 1212, 1216, 1236, 1278, 1457, 1458, 1459, 1460, 1461, 1462, 1463, 1473, 1474, 1476, 1484, 1489, 1510, 1515, 1531, 1540, 1541, 1542, 1543, 1581, 1586, 1593, 1595, 1596, 1598, 1599, 1606, 1608, 1609, 1616, 1619, 1623, 1625, 1628, 1634, 1642, 1643, 1645, 1649, 1675, 1685, 1702, 1704, 1705, 1706, 1721, 1729, 1732, 1734, 1735, 1737, 1738, 1739, 1751, 1757, 1761, 1788, 1789, 1797, 1801, 1802, 1807, 1851, 1854, 1855, 1880, 1890, 1905, 1921, 1922, 1927, 1928, 1940