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Die Steigung wird aber am ",[568,741,742],{},"neuen"," Punkt ",[576,745,747,794],{"className":746},[579],[576,748,750],{"className":749},[583],[585,751,752],{"xmlns":587},[589,753,754,791],{},[592,755,756,758,775,777,789],{},[602,757,616],{"stretchy":615},[759,760,761,763],"msub",{},[598,762,619],{},[592,764,765,768,771],{},[598,766,767],{},"n",[602,769,770],{},"+",[772,773,774],"mn",{},"1",[602,776,623],{"separator":622},[759,778,779,781],{},[598,780,600],{},[592,782,783,785,787],{},[598,784,767],{},[602,786,770],{},[772,788,774],{},[602,790,628],{"stretchy":615},[630,792,793],{"encoding":632},"(t_{n+1}, y_{n+1})",[576,795,797],{"className":796,"ariaHidden":622},[637],[576,798,800,803,806,862,865,868,918],{"className":799},[641],[576,801],{"className":802,"style":712},[645],[576,804,616],{"className":805},[720],[576,807,809,812],{"className":808},[650],[576,810,619],{"className":811},[650,654],[576,813,815],{"className":814},[659],[576,816,819,853],{"className":817},[663,818],"vlist-t2",[576,820,822,848],{"className":821},[667],[576,823,826],{"className":824,"style":825},[671],"height:0.3011em;",[576,827,829,832],{"style":828},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[576,830],{"className":831,"style":680},[679],[576,833,835],{"className":834},[684,685,686,687],[576,836,838,841,845],{"className":837},[650,687],[576,839,767],{"className":840},[650,654,687],[576,842,770],{"className":843},[844,687],"mbin",[576,846,774],{"className":847},[650,687],[576,849,852],{"className":850},[851],"vlist-s","​",[576,854,856],{"className":855},[667],[576,857,860],{"className":858,"style":859},[671],"height:0.2083em;",[576,861],{},[576,863,623],{"className":864},[727],[576,866],{"className":867,"style":731},[697],[576,869,871,874],{"className":870},[650],[576,872,600],{"className":873,"style":655},[650,654],[576,875,877],{"className":876},[659],[576,878,880,910],{"className":879},[663,818],[576,881,883,907],{"className":882},[667],[576,884,886],{"className":885,"style":825},[671],[576,887,889,892],{"style":888},"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;",[576,890],{"className":891,"style":680},[679],[576,893,895],{"className":894},[684,685,686,687],[576,896,898,901,904],{"className":897},[650,687],[576,899,767],{"className":900},[650,654,687],[576,902,770],{"className":903},[844,687],[576,905,774],{"className":906},[650,687],[576,908,852],{"className":909},[851],[576,911,913],{"className":912},[667],[576,914,916],{"className":915,"style":859},[671],[576,917],{},[576,919,628],{"className":920},[738]," ausgewertet. Da ",[576,923,925,949],{"className":924},[579],[576,926,928],{"className":927},[583],[585,929,930],{"xmlns":587},[589,931,932,946],{},[592,933,934],{},[759,935,936,938],{},[598,937,600],{},[592,939,940,942,944],{},[598,941,767],{},[602,943,770],{},[772,945,774],{},[630,947,948],{"encoding":632},"y_{n+1}",[576,950,952],{"className":951,"ariaHidden":622},[637],[576,953,955,959],{"className":954},[641],[576,956],{"className":957,"style":958},[645],"height:0.6389em;vertical-align:-0.2083em;",[576,960,962,965],{"className":961},[650],[576,963,600],{"className":964,"style":655},[650,654],[576,966,968],{"className":967},[659],[576,969,971,1000],{"className":970},[663,818],[576,972,974,997],{"className":973},[667],[576,975,977],{"className":976,"style":825},[671],[576,978,979,982],{"style":888},[576,980],{"className":981,"style":680},[679],[576,983,985],{"className":984},[684,685,686,687],[576,986,988,991,994],{"className":987},[650,687],[576,989,767],{"className":990},[650,654,687],[576,992,770],{"className":993},[844,687],[576,995,774],{"className":996},[650,687],[576,998,852],{"className":999},[851],[576,1001,1003],{"className":1002},[667],[576,1004,1006],{"className":1005,"style":859},[671],[576,1007],{}," auf beiden Seiten steht, muss man in jedem Schritt eine Gleichung lösen. Bei nichtlinearem ",[576,1010,1012,1025],{"className":1011},[579],[576,1013,1015],{"className":1014},[583],[585,1016,1017],{"xmlns":587},[589,1018,1019,1023],{},[592,1020,1021],{},[598,1022,612],{},[630,1024,612],{"encoding":632},[576,1026,1028],{"className":1027,"ariaHidden":622},[637],[576,1029,1031,1035],{"className":1030},[641],[576,1032],{"className":1033,"style":1034},[645],"height:0.8889em;vertical-align:-0.1944em;",[576,1036,612],{"className":1037,"style":716},[650,654]," geschieht das meist mit dem Newton-Verfahren, Startwert ist ",[576,1040,1042,1060],{"className":1041},[579],[576,1043,1045],{"className":1044},[583],[585,1046,1047],{"xmlns":587},[589,1048,1049,1057],{},[592,1050,1051],{},[759,1052,1053,1055],{},[598,1054,600],{},[598,1056,767],{},[630,1058,1059],{"encoding":632},"y_n",[576,1061,1063],{"className":1062,"ariaHidden":622},[637],[576,1064,1066,1070],{"className":1065},[641],[576,1067],{"className":1068,"style":1069},[645],"height:0.625em;vertical-align:-0.1944em;",[576,1071,1073,1076],{"className":1072},[650],[576,1074,600],{"className":1075,"style":655},[650,654],[576,1077,1079],{"className":1078},[659],[576,1080,1082,1103],{"className":1081},[663,818],[576,1083,1085,1100],{"className":1084},[667],[576,1086,1089],{"className":1087,"style":1088},[671],"height:0.1514em;",[576,1090,1091,1094],{"style":888},[576,1092],{"className":1093,"style":680},[679],[576,1095,1097],{"className":1096},[684,685,686,687],[576,1098,767],{"className":1099},[650,654,687],[576,1101,852],{"className":1102},[851],[576,1104,1106],{"className":1105},[667],[576,1107,1110],{"className":1108,"style":1109},[671],"height:0.15em;",[576,1111],{},".",[564,1114,1115],{},"Eigenschaften:",[1117,1118,1119,1126,1593],"ul",{},[1120,1121,1122,1125],"li",{},[568,1123,1124],{},"Konvergenzordnung 1",", wie beim expliziten Euler-Verfahren.",[1120,1127,1128,1131,1132,1222,1223,1278,1279,1530,1531,1534,1535,1563,1564,1592],{},[568,1129,1130],{},"A-stabil",": Für ",[576,1133,1135,1160],{"className":1134},[579],[576,1136,1138],{"className":1137},[583],[585,1139,1140],{"xmlns":587},[589,1141,1142,1157],{},[592,1143,1144,1150,1152,1155],{},[595,1145,1146,1148],{},[598,1147,600],{},[602,1149,606],{"mathvariant":604,"lspace":605,"rspace":605},[602,1151,609],{},[598,1153,1154],{},"λ",[598,1156,600],{},[630,1158,1159],{"encoding":632},"y' = \\lambda y",[576,1161,1163,1210],{"className":1162,"ariaHidden":622},[637],[576,1164,1166,1169,1201,1204,1207],{"className":1165},[641],[576,1167],{"className":1168,"style":646},[645],[576,1170,1172,1175],{"className":1171},[650],[576,1173,600],{"className":1174,"style":655},[650,654],[576,1176,1178],{"className":1177},[659],[576,1179,1181],{"className":1180},[663],[576,1182,1184],{"className":1183},[667],[576,1185,1187],{"className":1186,"style":672},[671],[576,1188,1189,1192],{"style":675},[576,1190],{"className":1191,"style":680},[679],[576,1193,1195],{"className":1194},[684,685,686,687],[576,1196,1198],{"className":1197},[650,687],[576,1199,606],{"className":1200},[650,687],[576,1202],{"className":1203,"style":698},[697],[576,1205,609],{"className":1206},[702],[576,1208],{"className":1209,"style":698},[697],[576,1211,1213,1216,1219],{"className":1212},[641],[576,1214],{"className":1215,"style":1034},[645],[576,1217,1154],{"className":1218},[650,654],[576,1220,600],{"className":1221,"style":655},[650,654]," mit ",[576,1224,1226,1246],{"className":1225},[579],[576,1227,1229],{"className":1228},[583],[585,1230,1231],{"xmlns":587},[589,1232,1233,1243],{},[592,1234,1235,1237,1240],{},[598,1236,1154],{},[602,1238,1239],{},"\u003C",[772,1241,1242],{},"0",[630,1244,1245],{"encoding":632},"\\lambda \u003C 0",[576,1247,1249,1268],{"className":1248,"ariaHidden":622},[637],[576,1250,1252,1256,1259,1262,1265],{"className":1251},[641],[576,1253],{"className":1254,"style":1255},[645],"height:0.7335em;vertical-align:-0.0391em;",[576,1257,1154],{"className":1258},[650,654],[576,1260],{"className":1261,"style":698},[697],[576,1263,1239],{"className":1264},[702],[576,1266],{"className":1267,"style":698},[697],[576,1269,1271,1275],{"className":1270},[641],[576,1272],{"className":1273,"style":1274},[645],"height:0.6444em;",[576,1276,1242],{"className":1277},[650]," ist ",[576,1280,1282,1329],{"className":1281},[579],[576,1283,1285],{"className":1284},[583],[585,1286,1287],{"xmlns":587},[589,1288,1289,1326],{},[592,1290,1291,1303,1305],{},[759,1292,1293,1295],{},[598,1294,600],{},[592,1296,1297,1299,1301],{},[598,1298,767],{},[602,1300,770],{},[772,1302,774],{},[602,1304,609],{},[1306,1307,1308,1314],"mfrac",{},[759,1309,1310,1312],{},[598,1311,600],{},[598,1313,767],{},[592,1315,1316,1318,1321,1324],{},[772,1317,774],{},[602,1319,1320],{},"−",[598,1322,1323],{},"h",[598,1325,1154],{},[630,1327,1328],{"encoding":632},"y_{n+1} = \\frac{y_n}{1 - h\\lambda}",[576,1330,1332,1396],{"className":1331,"ariaHidden":622},[637],[576,1333,1335,1338,1387,1390,1393],{"className":1334},[641],[576,1336],{"className":1337,"style":958},[645],[576,1339,1341,1344],{"className":1340},[650],[576,1342,600],{"className":1343,"style":655},[650,654],[576,1345,1347],{"className":1346},[659],[576,1348,1350,1379],{"className":1349},[663,818],[576,1351,1353,1376],{"className":1352},[667],[576,1354,1356],{"className":1355,"style":825},[671],[576,1357,1358,1361],{"style":888},[576,1359],{"className":1360,"style":680},[679],[576,1362,1364],{"className":1363},[684,685,686,687],[576,1365,1367,1370,1373],{"className":1366},[650,687],[576,1368,767],{"className":1369},[650,654,687],[576,1371,770],{"className":1372},[844,687],[576,1374,774],{"className":1375},[650,687],[576,1377,852],{"className":1378},[851],[576,1380,1382],{"className":1381},[667],[576,1383,1385],{"className":1384,"style":859},[671],[576,1386],{},[576,1388],{"className":1389,"style":698},[697],[576,1391,609],{"className":1392},[702],[576,1394],{"className":1395,"style":698},[697],[576,1397,1399,1403],{"className":1398},[641],[576,1400],{"className":1401,"style":1402},[645],"height:1.1508em;vertical-align:-0.4033em;",[576,1404,1406,1410,1527],{"className":1405},[650],[576,1407],{"className":1408},[720,1409],"nulldelimiter",[576,1411,1413],{"className":1412},[1306],[576,1414,1416,1518],{"className":1415},[663,818],[576,1417,1419,1515],{"className":1418},[667],[576,1420,1423,1446,1457],{"className":1421,"style":1422},[671],"height:0.7475em;",[576,1424,1426,1430],{"style":1425},"top:-2.655em;",[576,1427],{"className":1428,"style":1429},[679],"height:3em;",[576,1431,1433],{"className":1432},[684,685,686,687],[576,1434,1436,1439,1442],{"className":1435},[650,687],[576,1437,774],{"className":1438},[650,687],[576,1440,1320],{"className":1441},[844,687],[576,1443,1445],{"className":1444},[650,654,687],"hλ",[576,1447,1449,1452],{"style":1448},"top:-3.23em;",[576,1450],{"className":1451,"style":1429},[679],[576,1453],{"className":1454,"style":1456},[1455],"frac-line","border-bottom-width:0.04em;",[576,1458,1460,1463],{"style":1459},"top:-3.4461em;",[576,1461],{"className":1462,"style":1429},[679],[576,1464,1466],{"className":1465},[684,685,686,687],[576,1467,1469],{"className":1468},[650,687],[576,1470,1472,1475],{"className":1471},[650,687],[576,1473,600],{"className":1474,"style":655},[650,654,687],[576,1476,1478],{"className":1477},[659],[576,1479,1481,1506],{"className":1480},[663,818],[576,1482,1484,1503],{"className":1483},[667],[576,1485,1488],{"className":1486,"style":1487},[671],"height:0.1645em;",[576,1489,1491,1495],{"style":1490},"top:-2.357em;margin-left:-0.0359em;margin-right:0.0714em;",[576,1492],{"className":1493,"style":1494},[679],"height:2.5em;",[576,1496,1500],{"className":1497},[684,1498,1499,687],"reset-size3","size1",[576,1501,767],{"className":1502},[650,654,687],[576,1504,852],{"className":1505},[851],[576,1507,1509],{"className":1508},[667],[576,1510,1513],{"className":1511,"style":1512},[671],"height:0.143em;",[576,1514],{},[576,1516,852],{"className":1517},[851],[576,1519,1521],{"className":1520},[667],[576,1522,1525],{"className":1523,"style":1524},[671],"height:0.4033em;",[576,1526],{},[576,1528],{"className":1529},[738,1409],". Der Faktor liegt für ",[568,1532,1533],{},"jede"," Schrittweite zwischen ",[576,1536,1538,1551],{"className":1537},[579],[576,1539,1541],{"className":1540},[583],[585,1542,1543],{"xmlns":587},[589,1544,1545,1549],{},[592,1546,1547],{},[772,1548,1242],{},[630,1550,1242],{"encoding":632},[576,1552,1554],{"className":1553,"ariaHidden":622},[637],[576,1555,1557,1560],{"className":1556},[641],[576,1558],{"className":1559,"style":1274},[645],[576,1561,1242],{"className":1562},[650]," und ",[576,1565,1567,1580],{"className":1566},[579],[576,1568,1570],{"className":1569},[583],[585,1571,1572],{"xmlns":587},[589,1573,1574,1578],{},[592,1575,1576],{},[772,1577,774],{},[630,1579,774],{"encoding":632},[576,1581,1583],{"className":1582,"ariaHidden":622},[637],[576,1584,1586,1589],{"className":1585},[641],[576,1587],{"className":1588,"style":1274},[645],[576,1590,774],{"className":1591},[650],". Die Näherung klingt also immer ab, wie die echte Lösung.",[1120,1594,1595],{},"Pro Schritt teurer als explizite Verfahren, weil eine Gleichung gelöst werden muss.",[564,1597,1598,1599,1602],{},"Diese Stabilität macht das Verfahren wertvoll für ",[568,1600,1601],{},"steife"," Differentialgleichungen. Das sind Gleichungen, in denen sehr schnell und sehr langsam abklingende Anteile zusammenkommen, etwa in der chemischen Kinetik oder bei elektrischen Schaltungen. Explizite Verfahren brauchen dort winzige Schritte, nur um stabil zu bleiben.",[559,1604,1606],{"id":1605},"beispiele","Beispiele",[564,1608,1609,1705,1706,1772,1773,1849,1850,1909,1910,1972,1973,2076,2077,2187],{},[576,1610,1612,1639],{"className":1611},[579],[576,1613,1615],{"className":1614},[583],[585,1616,1617],{"xmlns":587},[589,1618,1619,1636],{},[592,1620,1621,1627,1629,1631,1634],{},[595,1622,1623,1625],{},[598,1624,600],{},[602,1626,606],{"mathvariant":604,"lspace":605,"rspace":605},[602,1628,609],{},[602,1630,1320],{},[772,1632,1633],{},"2",[598,1635,600],{},[630,1637,1638],{"encoding":632},"y' = -2y",[576,1640,1642,1689],{"className":1641,"ariaHidden":622},[637],[576,1643,1645,1648,1680,1683,1686],{"className":1644},[641],[576,1646],{"className":1647,"style":646},[645],[576,1649,1651,1654],{"className":1650},[650],[576,1652,600],{"className":1653,"style":655},[650,654],[576,1655,1657],{"className":1656},[659],[576,1658,1660],{"className":1659},[663],[576,1661,1663],{"className":1662},[667],[576,1664,1666],{"className":1665,"style":672},[671],[576,1667,1668,1671],{"style":675},[576,1669],{"className":1670,"style":680},[679],[576,1672,1674],{"className":1673},[684,685,686,687],[576,1675,1677],{"className":1676},[650,687],[576,1678,606],{"className":1679},[650,687],[576,1681],{"className":1682,"style":698},[697],[576,1684,609],{"className":1685},[702],[576,1687],{"className":1688,"style":698},[697],[576,1690,1692,1696,1699,1702],{"className":1691},[641],[576,1693],{"className":1694,"style":1695},[645],"height:0.8389em;vertical-align:-0.1944em;",[576,1697,1320],{"className":1698},[650],[576,1700,1633],{"className":1701},[650],[576,1703,600],{"className":1704,"style":655},[650,654],", ",[576,1707,1709,1733],{"className":1708},[579],[576,1710,1712],{"className":1711},[583],[585,1713,1714],{"xmlns":587},[589,1715,1716,1730],{},[592,1717,1718,1720,1722,1724,1726,1728],{},[598,1719,600],{},[602,1721,616],{"stretchy":615},[772,1723,1242],{},[602,1725,628],{"stretchy":615},[602,1727,609],{},[772,1729,774],{},[630,1731,1732],{"encoding":632},"y(0) = 1",[576,1734,1736,1763],{"className":1735,"ariaHidden":622},[637],[576,1737,1739,1742,1745,1748,1751,1754,1757,1760],{"className":1738},[641],[576,1740],{"className":1741,"style":712},[645],[576,1743,600],{"className":1744,"style":655},[650,654],[576,1746,616],{"className":1747},[720],[576,1749,1242],{"className":1750},[650],[576,1752,628],{"className":1753},[738],[576,1755],{"className":1756,"style":698},[697],[576,1758,609],{"className":1759},[702],[576,1761],{"className":1762,"style":698},[697],[576,1764,1766,1769],{"className":1765},[641],[576,1767],{"className":1768,"style":1274},[645],[576,1770,774],{"className":1771},[650],", exakte Lösung ",[576,1774,1776,1801],{"className":1775},[579],[576,1777,1779],{"className":1778},[583],[585,1780,1781],{"xmlns":587},[589,1782,1783,1798],{},[592,1784,1785],{},[595,1786,1787,1790],{},[598,1788,1789],{},"e",[592,1791,1792,1794,1796],{},[602,1793,1320],{},[772,1795,1633],{},[598,1797,619],{},[630,1799,1800],{"encoding":632},"e^{-2t}",[576,1802,1804],{"className":1803,"ariaHidden":622},[637],[576,1805,1807,1811],{"className":1806},[641],[576,1808],{"className":1809,"style":1810},[645],"height:0.8141em;",[576,1812,1814,1817],{"className":1813},[650],[576,1815,1789],{"className":1816},[650,654],[576,1818,1820],{"className":1819},[659],[576,1821,1823],{"className":1822},[663],[576,1824,1826],{"className":1825},[667],[576,1827,1829],{"className":1828,"style":1810},[671],[576,1830,1831,1834],{"style":675},[576,1832],{"className":1833,"style":680},[679],[576,1835,1837],{"className":1836},[684,685,686,687],[576,1838,1840,1843,1846],{"className":1839},[650,687],[576,1841,1320],{"className":1842},[650,687],[576,1844,1633],{"className":1845},[650,687],[576,1847,619],{"className":1848},[650,654,687],". Jeder Schritt teilt durch ",[576,1851,1853,1873],{"className":1852},[579],[576,1854,1856],{"className":1855},[583],[585,1857,1858],{"xmlns":587},[589,1859,1860,1870],{},[592,1861,1862,1864,1866,1868],{},[772,1863,774],{},[602,1865,770],{},[772,1867,1633],{},[598,1869,1323],{},[630,1871,1872],{"encoding":632},"1 + 2h",[576,1874,1876,1896],{"className":1875,"ariaHidden":622},[637],[576,1877,1879,1883,1886,1890,1893],{"className":1878},[641],[576,1880],{"className":1881,"style":1882},[645],"height:0.7278em;vertical-align:-0.0833em;",[576,1884,774],{"className":1885},[650],[576,1887],{"className":1888,"style":1889},[697],"margin-right:0.2222em;",[576,1891,770],{"className":1892},[844],[576,1894],{"className":1895,"style":1889},[697],[576,1897,1899,1903,1906],{"className":1898},[641],[576,1900],{"className":1901,"style":1902},[645],"height:0.6944em;",[576,1904,1633],{"className":1905},[650],[576,1907,1323],{"className":1908},[650,654],". Mit ",[576,1911,1913,1932],{"className":1912},[579],[576,1914,1916],{"className":1915},[583],[585,1917,1918],{"xmlns":587},[589,1919,1920,1929],{},[592,1921,1922,1924,1926],{},[598,1923,1323],{},[602,1925,609],{},[772,1927,1928],{},"1,5",[630,1930,1931],{"encoding":632},"h = 1{,}5",[576,1933,1935,1953],{"className":1934,"ariaHidden":622},[637],[576,1936,1938,1941,1944,1947,1950],{"className":1937},[641],[576,1939],{"className":1940,"style":1902},[645],[576,1942,1323],{"className":1943},[650,654],[576,1945],{"className":1946,"style":698},[697],[576,1948,609],{"className":1949},[702],[576,1951],{"className":1952,"style":698},[697],[576,1954,1956,1959,1962,1968],{"className":1955},[641],[576,1957],{"className":1958,"style":1695},[645],[576,1960,774],{"className":1961},[650],[576,1963,1965],{"className":1964},[650],[576,1966,623],{"className":1967},[727],[576,1969,1971],{"className":1970},[650],"5"," ist der Faktor ",[576,1974,1976,1995],{"className":1975},[579],[576,1977,1979],{"className":1978},[583],[585,1980,1981],{"xmlns":587},[589,1982,1983,1992],{},[592,1984,1985],{},[1306,1986,1987,1989],{},[772,1988,774],{},[772,1990,1991],{},"4",[630,1993,1994],{"encoding":632},"\\frac{1}{4}",[576,1996,1998],{"className":1997,"ariaHidden":622},[637],[576,1999,2001,2005],{"className":2000},[641],[576,2002],{"className":2003,"style":2004},[645],"height:1.1901em;vertical-align:-0.345em;",[576,2006,2008,2011,2073],{"className":2007},[650],[576,2009],{"className":2010},[720,1409],[576,2012,2014],{"className":2013},[1306],[576,2015,2017,2064],{"className":2016},[663,818],[576,2018,2020,2061],{"className":2019},[667],[576,2021,2024,2038,2046],{"className":2022,"style":2023},[671],"height:0.8451em;",[576,2025,2026,2029],{"style":1425},[576,2027],{"className":2028,"style":1429},[679],[576,2030,2032],{"className":2031},[684,685,686,687],[576,2033,2035],{"className":2034},[650,687],[576,2036,1991],{"className":2037},[650,687],[576,2039,2040,2043],{"style":1448},[576,2041],{"className":2042,"style":1429},[679],[576,2044],{"className":2045,"style":1456},[1455],[576,2047,2049,2052],{"style":2048},"top:-3.394em;",[576,2050],{"className":2051,"style":1429},[679],[576,2053,2055],{"className":2054},[684,685,686,687],[576,2056,2058],{"className":2057},[650,687],[576,2059,774],{"className":2060},[650,687],[576,2062,852],{"className":2063},[851],[576,2065,2067],{"className":2066},[667],[576,2068,2071],{"className":2069,"style":2070},[671],"height:0.345em;",[576,2072],{},[576,2074],{"className":2075},[738,1409],", die Näherung fällt monoton: ",[576,2078,2080,2118],{"className":2079},[579],[576,2081,2083],{"className":2082},[583],[585,2084,2085],{"xmlns":587},[589,2086,2087,2115],{},[592,2088,2089,2091,2094,2098,2101,2103,2105,2108,2110,2112],{},[772,2090,774],{},[602,2092,2093],{"separator":622},";",[2095,2096,2097],"mtext",{}," ",[772,2099,2100],{},"0,25",[602,2102,2093],{"separator":622},[2095,2104,2097],{},[772,2106,2107],{},"0,0625",[602,2109,2093],{"separator":622},[2095,2111,2097],{},[602,2113,2114],{},"…",[630,2116,2117],{"encoding":632},"1;\\ 0{,}25;\\ 0{,}0625;\\ \\ldots",[576,2119,2121],{"className":2120,"ariaHidden":622},[637],[576,2122,2124,2127,2130,2133,2136,2139,2142,2148,2152,2155,2158,2161,2164,2170,2174,2177,2180,2183],{"className":2123},[641],[576,2125],{"className":2126,"style":1695},[645],[576,2128,774],{"className":2129},[650],[576,2131,2093],{"className":2132},[727],[576,2134,2097],{"className":2135},[697],[576,2137],{"className":2138,"style":731},[697],[576,2140,1242],{"className":2141},[650],[576,2143,2145],{"className":2144},[650],[576,2146,623],{"className":2147},[727],[576,2149,2151],{"className":2150},[650],"25",[576,2153,2093],{"className":2154},[727],[576,2156,2097],{"className":2157},[697],[576,2159],{"className":2160,"style":731},[697],[576,2162,1242],{"className":2163},[650],[576,2165,2167],{"className":2166},[650],[576,2168,623],{"className":2169},[727],[576,2171,2173],{"className":2172},[650],"0625",[576,2175,2093],{"className":2176},[727],[576,2178,2097],{"className":2179},[697],[576,2181],{"className":2182,"style":731},[697],[576,2184,2114],{"className":2185},[2186],"minner"," Das explizite Verfahren springt bei dieser Schrittweite zwischen wachsenden positiven und negativen Werten.",[564,2189,2190,2191,1278,2252,2459,2460,2639,2640,2742],{},"Mit ",[576,2192,2194,2213],{"className":2193},[579],[576,2195,2197],{"className":2196},[583],[585,2198,2199],{"xmlns":587},[589,2200,2201,2210],{},[592,2202,2203,2205,2207],{},[598,2204,1323],{},[602,2206,609],{},[772,2208,2209],{},"0,1",[630,2211,2212],{"encoding":632},"h = 0{,}1",[576,2214,2216,2234],{"className":2215,"ariaHidden":622},[637],[576,2217,2219,2222,2225,2228,2231],{"className":2218},[641],[576,2220],{"className":2221,"style":1902},[645],[576,2223,1323],{"className":2224},[650,654],[576,2226],{"className":2227,"style":698},[697],[576,2229,609],{"className":2230},[702],[576,2232],{"className":2233,"style":698},[697],[576,2235,2237,2240,2243,2249],{"className":2236},[641],[576,2238],{"className":2239,"style":1695},[645],[576,2241,1242],{"className":2242},[650],[576,2244,2246],{"className":2245},[650],[576,2247,623],{"className":2248},[727],[576,2250,774],{"className":2251},[650],[576,2253,2255,2288],{"className":2254},[579],[576,2256,2258],{"className":2257},[583],[585,2259,2260],{"xmlns":587},[589,2261,2262,2285],{},[592,2263,2264,2270,2272,2279,2282],{},[759,2265,2266,2268],{},[598,2267,600],{},[772,2269,774],{},[602,2271,609],{},[1306,2273,2274,2276],{},[772,2275,774],{},[772,2277,2278],{},"1,2",[602,2280,2281],{},"≈",[772,2283,2284],{},"0,8333",[630,2286,2287],{"encoding":632},"y_1 = \\frac{1}{1{,}2} \\approx 0{,}8333",[576,2289,2291,2346,2440],{"className":2290,"ariaHidden":622},[637],[576,2292,2294,2297,2337,2340,2343],{"className":2293},[641],[576,2295],{"className":2296,"style":1069},[645],[576,2298,2300,2303],{"className":2299},[650],[576,2301,600],{"className":2302,"style":655},[650,654],[576,2304,2306],{"className":2305},[659],[576,2307,2309,2329],{"className":2308},[663,818],[576,2310,2312,2326],{"className":2311},[667],[576,2313,2315],{"className":2314,"style":825},[671],[576,2316,2317,2320],{"style":888},[576,2318],{"className":2319,"style":680},[679],[576,2321,2323],{"className":2322},[684,685,686,687],[576,2324,774],{"className":2325},[650,687],[576,2327,852],{"className":2328},[851],[576,2330,2332],{"className":2331},[667],[576,2333,2335],{"className":2334,"style":1109},[671],[576,2336],{},[576,2338],{"className":2339,"style":698},[697],[576,2341,609],{"className":2342},[702],[576,2344],{"className":2345,"style":698},[697],[576,2347,2349,2353,2431,2434,2437],{"className":2348},[641],[576,2350],{"className":2351,"style":2352},[645],"height:1.3262em;vertical-align:-0.4811em;",[576,2354,2356,2359,2428],{"className":2355},[650],[576,2357],{"className":2358},[720,1409],[576,2360,2362],{"className":2361},[1306],[576,2363,2365,2419],{"className":2364},[663,818],[576,2366,2368,2416],{"className":2367},[667],[576,2369,2371,2394,2402],{"className":2370,"style":2023},[671],[576,2372,2373,2376],{"style":1425},[576,2374],{"className":2375,"style":1429},[679],[576,2377,2379],{"className":2378},[684,685,686,687],[576,2380,2382,2385,2391],{"className":2381},[650,687],[576,2383,774],{"className":2384},[650,687],[576,2386,2388],{"className":2387},[650,687],[576,2389,623],{"className":2390},[727,687],[576,2392,1633],{"className":2393},[650,687],[576,2395,2396,2399],{"style":1448},[576,2397],{"className":2398,"style":1429},[679],[576,2400],{"className":2401,"style":1456},[1455],[576,2403,2404,2407],{"style":2048},[576,2405],{"className":2406,"style":1429},[679],[576,2408,2410],{"className":2409},[684,685,686,687],[576,2411,2413],{"className":2412},[650,687],[576,2414,774],{"className":2415},[650,687],[576,2417,852],{"className":2418},[851],[576,2420,2422],{"className":2421},[667],[576,2423,2426],{"className":2424,"style":2425},[671],"height:0.4811em;",[576,2427],{},[576,2429],{"className":2430},[738,1409],[576,2432],{"className":2433,"style":698},[697],[576,2435,2281],{"className":2436},[702],[576,2438],{"className":2439,"style":698},[697],[576,2441,2443,2446,2449,2455],{"className":2442},[641],[576,2444],{"className":2445,"style":1695},[645],[576,2447,1242],{"className":2448},[650],[576,2450,2452],{"className":2451},[650],[576,2453,623],{"className":2454},[727],[576,2456,2458],{"className":2457},[650],"8333",", nach zehn Schritten ",[576,2461,2463,2499],{"className":2462},[579],[576,2464,2466],{"className":2465},[583],[585,2467,2468],{"xmlns":587},[589,2469,2470,2496],{},[592,2471,2472,2479,2481,2491,2493],{},[759,2473,2474,2476],{},[598,2475,600],{},[772,2477,2478],{},"10",[602,2480,609],{},[595,2482,2483,2485],{},[772,2484,2278],{},[592,2486,2487,2489],{},[602,2488,1320],{},[772,2490,2478],{},[602,2492,2281],{},[772,2494,2495],{},"0,1615",[630,2497,2498],{"encoding":632},"y_{10} = 1{,}2^{-10} \\approx 0{,}1615",[576,2500,2502,2560,2620],{"className":2501,"ariaHidden":622},[637],[576,2503,2505,2508,2551,2554,2557],{"className":2504},[641],[576,2506],{"className":2507,"style":1069},[645],[576,2509,2511,2514],{"className":2510},[650],[576,2512,600],{"className":2513,"style":655},[650,654],[576,2515,2517],{"className":2516},[659],[576,2518,2520,2543],{"className":2519},[663,818],[576,2521,2523,2540],{"className":2522},[667],[576,2524,2526],{"className":2525,"style":825},[671],[576,2527,2528,2531],{"style":888},[576,2529],{"className":2530,"style":680},[679],[576,2532,2534],{"className":2533},[684,685,686,687],[576,2535,2537],{"className":2536},[650,687],[576,2538,2478],{"className":2539},[650,687],[576,2541,852],{"className":2542},[851],[576,2544,2546],{"className":2545},[667],[576,2547,2549],{"className":2548,"style":1109},[671],[576,2550],{},[576,2552],{"className":2553,"style":698},[697],[576,2555,609],{"className":2556},[702],[576,2558],{"className":2559,"style":698},[697],[576,2561,2563,2567,2570,2576,2611,2614,2617],{"className":2562},[641],[576,2564],{"className":2565,"style":2566},[645],"height:1.0085em;vertical-align:-0.1944em;",[576,2568,774],{"className":2569},[650],[576,2571,2573],{"className":2572},[650],[576,2574,623],{"className":2575},[727],[576,2577,2579,2582],{"className":2578},[650],[576,2580,1633],{"className":2581},[650],[576,2583,2585],{"className":2584},[659],[576,2586,2588],{"className":2587},[663],[576,2589,2591],{"className":2590},[667],[576,2592,2594],{"className":2593,"style":1810},[671],[576,2595,2596,2599],{"style":675},[576,2597],{"className":2598,"style":680},[679],[576,2600,2602],{"className":2601},[684,685,686,687],[576,2603,2605,2608],{"className":2604},[650,687],[576,2606,1320],{"className":2607},[650,687],[576,2609,2478],{"className":2610},[650,687],[576,2612],{"className":2613,"style":698},[697],[576,2615,2281],{"className":2616},[702],[576,2618],{"className":2619,"style":698},[697],[576,2621,2623,2626,2629,2635],{"className":2622},[641],[576,2624],{"className":2625,"style":1695},[645],[576,2627,1242],{"className":2628},[650],[576,2630,2632],{"className":2631},[650],[576,2633,623],{"className":2634},[727],[576,2636,2638],{"className":2637},[650],"1615",". Exakt ist ",[576,2641,2643,2670],{"className":2642},[579],[576,2644,2646],{"className":2645},[583],[585,2647,2648],{"xmlns":587},[589,2649,2650,2667],{},[592,2651,2652,2662,2664],{},[595,2653,2654,2656],{},[598,2655,1789],{},[592,2657,2658,2660],{},[602,2659,1320],{},[772,2661,1633],{},[602,2663,2281],{},[772,2665,2666],{},"0,1353",[630,2668,2669],{"encoding":632},"e^{-2} \\approx 0{,}1353",[576,2671,2673,2723],{"className":2672,"ariaHidden":622},[637],[576,2674,2676,2679,2714,2717,2720],{"className":2675},[641],[576,2677],{"className":2678,"style":1810},[645],[576,2680,2682,2685],{"className":2681},[650],[576,2683,1789],{"className":2684},[650,654],[576,2686,2688],{"className":2687},[659],[576,2689,2691],{"className":2690},[663],[576,2692,2694],{"className":2693},[667],[576,2695,2697],{"className":2696,"style":1810},[671],[576,2698,2699,2702],{"style":675},[576,2700],{"className":2701,"style":680},[679],[576,2703,2705],{"className":2704},[684,685,686,687],[576,2706,2708,2711],{"className":2707},[650,687],[576,2709,1320],{"className":2710},[650,687],[576,2712,1633],{"className":2713},[650,687],[576,2715],{"className":2716,"style":698},[697],[576,2718,2281],{"className":2719},[702],[576,2721],{"className":2722,"style":698},[697],[576,2724,2726,2729,2732,2738],{"className":2725},[641],[576,2727],{"className":2728,"style":1695},[645],[576,2730,1242],{"className":2731},[650],[576,2733,2735],{"className":2734},[650],[576,2736,623],{"className":2737},[727],[576,2739,2741],{"className":2740},[650],"1353",". Der Fehler ist ähnlich groß wie beim expliziten Verfahren, nur mit anderem Vorzeichen.",[564,2744,2745,1705,2864,2929,2930,2991],{},[576,2746,2748,2776],{"className":2747},[579],[576,2749,2751],{"className":2750},[583],[585,2752,2753],{"xmlns":587},[589,2754,2755,2773],{},[592,2756,2757,2763,2765,2767],{},[595,2758,2759,2761],{},[598,2760,600],{},[602,2762,606],{"mathvariant":604,"lspace":605,"rspace":605},[602,2764,609],{},[602,2766,1320],{},[595,2768,2769,2771],{},[598,2770,600],{},[772,2772,1633],{},[630,2774,2775],{"encoding":632},"y' = -y^2",[576,2777,2779,2826],{"className":2778,"ariaHidden":622},[637],[576,2780,2782,2785,2817,2820,2823],{"className":2781},[641],[576,2783],{"className":2784,"style":646},[645],[576,2786,2788,2791],{"className":2787},[650],[576,2789,600],{"className":2790,"style":655},[650,654],[576,2792,2794],{"className":2793},[659],[576,2795,2797],{"className":2796},[663],[576,2798,2800],{"className":2799},[667],[576,2801,2803],{"className":2802,"style":672},[671],[576,2804,2805,2808],{"style":675},[576,2806],{"className":2807,"style":680},[679],[576,2809,2811],{"className":2810},[684,685,686,687],[576,2812,2814],{"className":2813},[650,687],[576,2815,606],{"className":2816},[650,687],[576,2818],{"className":2819,"style":698},[697],[576,2821,609],{"className":2822},[702],[576,2824],{"className":2825,"style":698},[697],[576,2827,2829,2832,2835],{"className":2828},[641],[576,2830],{"className":2831,"style":2566},[645],[576,2833,1320],{"className":2834},[650],[576,2836,2838,2841],{"className":2837},[650],[576,2839,600],{"className":2840,"style":655},[650,654],[576,2842,2844],{"className":2843},[659],[576,2845,2847],{"className":2846},[663],[576,2848,2850],{"className":2849},[667],[576,2851,2853],{"className":2852,"style":1810},[671],[576,2854,2855,2858],{"style":675},[576,2856],{"className":2857,"style":680},[679],[576,2859,2861],{"className":2860},[684,685,686,687],[576,2862,1633],{"className":2863},[650,687],[576,2865,2867,2890],{"className":2866},[579],[576,2868,2870],{"className":2869},[583],[585,2871,2872],{"xmlns":587},[589,2873,2874,2888],{},[592,2875,2876,2878,2880,2882,2884,2886],{},[598,2877,600],{},[602,2879,616],{"stretchy":615},[772,2881,1242],{},[602,2883,628],{"stretchy":615},[602,2885,609],{},[772,2887,774],{},[630,2889,1732],{"encoding":632},[576,2891,2893,2920],{"className":2892,"ariaHidden":622},[637],[576,2894,2896,2899,2902,2905,2908,2911,2914,2917],{"className":2895},[641],[576,2897],{"className":2898,"style":712},[645],[576,2900,600],{"className":2901,"style":655},[650,654],[576,2903,616],{"className":2904},[720],[576,2906,1242],{"className":2907},[650],[576,2909,628],{"className":2910},[738],[576,2912],{"className":2913,"style":698},[697],[576,2915,609],{"className":2916},[702],[576,2918],{"className":2919,"style":698},[697],[576,2921,2923,2926],{"className":2922},[641],[576,2924],{"className":2925,"style":1274},[645],[576,2927,774],{"className":2928},[650],", ein Schritt mit ",[576,2931,2933,2952],{"className":2932},[579],[576,2934,2936],{"className":2935},[583],[585,2937,2938],{"xmlns":587},[589,2939,2940,2949],{},[592,2941,2942,2944,2946],{},[598,2943,1323],{},[602,2945,609],{},[772,2947,2948],{},"0,5",[630,2950,2951],{"encoding":632},"h = 0{,}5",[576,2953,2955,2973],{"className":2954,"ariaHidden":622},[637],[576,2956,2958,2961,2964,2967,2970],{"className":2957},[641],[576,2959],{"className":2960,"style":1902},[645],[576,2962,1323],{"className":2963},[650,654],[576,2965],{"className":2966,"style":698},[697],[576,2968,609],{"className":2969},[702],[576,2971],{"className":2972,"style":698},[697],[576,2974,2976,2979,2982,2988],{"className":2975},[641],[576,2977],{"className":2978,"style":1695},[645],[576,2980,1242],{"className":2981},[650],[576,2983,2985],{"className":2984},[650],[576,2986,623],{"className":2987},[727],[576,2989,1971],{"className":2990},[650],". Die Schrittgleichung lautet",[576,2993,2996],{"className":2994},[2995],"katex-display",[576,2997,2999,3072],{"className":2998},[579],[576,3000,3002],{"className":3001},[583],[585,3003,3005],{"xmlns":587,"display":3004},"block",[589,3006,3007,3069],{},[592,3008,3009,3015,3017,3019,3021,3023,3026,3035,3038,3041,3043,3051,3053,3055,3061,3063,3065,3067],{},[759,3010,3011,3013],{},[598,3012,600],{},[772,3014,774],{},[602,3016,609],{},[772,3018,774],{},[602,3020,1320],{},[772,3022,2948],{},[2095,3024,3025],{}," ",[3027,3028,3029,3031,3033],"msubsup",{},[598,3030,600],{},[772,3032,774],{},[772,3034,1633],{},[697,3036],{"width":3037},"1em",[602,3039,3040],{},"⇔",[697,3042],{"width":3037},[3027,3044,3045,3047,3049],{},[598,3046,600],{},[772,3048,774],{},[772,3050,1633],{},[602,3052,770],{},[772,3054,1633],{},[759,3056,3057,3059],{},[598,3058,600],{},[772,3060,774],{},[602,3062,1320],{},[772,3064,1633],{},[602,3066,609],{},[772,3068,1242],{},[630,3070,3071],{"encoding":632},"y_1 = 1 - 0{,}5\\,y_1^2 \\quad\\Leftrightarrow\\quad y_1^2 + 2y_1 - 2 = 0",[576,3073,3075,3130,3148,3241,3307,3365,3383],{"className":3074,"ariaHidden":622},[637],[576,3076,3078,3081,3121,3124,3127],{"className":3077},[641],[576,3079],{"className":3080,"style":1069},[645],[576,3082,3084,3087],{"className":3083},[650],[576,3085,600],{"className":3086,"style":655},[650,654],[576,3088,3090],{"className":3089},[659],[576,3091,3093,3113],{"className":3092},[663,818],[576,3094,3096,3110],{"className":3095},[667],[576,3097,3099],{"className":3098,"style":825},[671],[576,3100,3101,3104],{"style":888},[576,3102],{"className":3103,"style":680},[679],[576,3105,3107],{"className":3106},[684,685,686,687],[576,3108,774],{"className":3109},[650,687],[576,3111,852],{"className":3112},[851],[576,3114,3116],{"className":3115},[667],[576,3117,3119],{"className":3118,"style":1109},[671],[576,3120],{},[576,3122],{"className":3123,"style":698},[697],[576,3125,609],{"className":3126},[702],[576,3128],{"className":3129,"style":698},[697],[576,3131,3133,3136,3139,3142,3145],{"className":3132},[641],[576,3134],{"className":3135,"style":1882},[645],[576,3137,774],{"className":3138},[650],[576,3140],{"className":3141,"style":1889},[697],[576,3143,1320],{"className":3144},[844],[576,3146],{"className":3147,"style":1889},[697],[576,3149,3151,3155,3158,3164,3167,3170,3225,3229,3232,3235,3238],{"className":3150},[641],[576,3152],{"className":3153,"style":3154},[645],"height:1.1111em;vertical-align:-0.247em;",[576,3156,1242],{"className":3157},[650],[576,3159,3161],{"className":3160},[650],[576,3162,623],{"className":3163},[727],[576,3165,1971],{"className":3166},[650],[576,3168],{"className":3169,"style":731},[697],[576,3171,3173,3176],{"className":3172},[650],[576,3174,600],{"className":3175,"style":655},[650,654],[576,3177,3179],{"className":3178},[659],[576,3180,3182,3216],{"className":3181},[663,818],[576,3183,3185,3213],{"className":3184},[667],[576,3186,3189,3201],{"className":3187,"style":3188},[671],"height:0.8641em;",[576,3190,3192,3195],{"style":3191},"top:-2.453em;margin-left:-0.0359em;margin-right:0.05em;",[576,3193],{"className":3194,"style":680},[679],[576,3196,3198],{"className":3197},[684,685,686,687],[576,3199,774],{"className":3200},[650,687],[576,3202,3204,3207],{"style":3203},"top:-3.113em;margin-right:0.05em;",[576,3205],{"className":3206,"style":680},[679],[576,3208,3210],{"className":3209},[684,685,686,687],[576,3211,1633],{"className":3212},[650,687],[576,3214,852],{"className":3215},[851],[576,3217,3219],{"className":3218},[667],[576,3220,3223],{"className":3221,"style":3222},[671],"height:0.247em;",[576,3224],{},[576,3226],{"className":3227,"style":3228},[697],"margin-right:1em;",[576,3230],{"className":3231,"style":698},[697],[576,3233,3040],{"className":3234},[702],[576,3236],{"className":3237,"style":3228},[697],[576,3239],{"className":3240,"style":698},[697],[576,3242,3244,3247,3298,3301,3304],{"className":3243},[641],[576,3245],{"className":3246,"style":3154},[645],[576,3248,3250,3253],{"className":3249},[650],[576,3251,600],{"className":3252,"style":655},[650,654],[576,3254,3256],{"className":3255},[659],[576,3257,3259,3290],{"className":3258},[663,818],[576,3260,3262,3287],{"className":3261},[667],[576,3263,3265,3276],{"className":3264,"style":3188},[671],[576,3266,3267,3270],{"style":3191},[576,3268],{"className":3269,"style":680},[679],[576,3271,3273],{"className":3272},[684,685,686,687],[576,3274,774],{"className":3275},[650,687],[576,3277,3278,3281],{"style":3203},[576,3279],{"className":3280,"style":680},[679],[576,3282,3284],{"className":3283},[684,685,686,687],[576,3285,1633],{"className":3286},[650,687],[576,3288,852],{"className":3289},[851],[576,3291,3293],{"className":3292},[667],[576,3294,3296],{"className":3295,"style":3222},[671],[576,3297],{},[576,3299],{"className":3300,"style":1889},[697],[576,3302,770],{"className":3303},[844],[576,3305],{"className":3306,"style":1889},[697],[576,3308,3310,3313,3316,3356,3359,3362],{"className":3309},[641],[576,3311],{"className":3312,"style":1695},[645],[576,3314,1633],{"className":3315},[650],[576,3317,3319,3322],{"className":3318},[650],[576,3320,600],{"className":3321,"style":655},[650,654],[576,3323,3325],{"className":3324},[659],[576,3326,3328,3348],{"className":3327},[663,818],[576,3329,3331,3345],{"className":3330},[667],[576,3332,3334],{"className":3333,"style":825},[671],[576,3335,3336,3339],{"style":888},[576,3337],{"className":3338,"style":680},[679],[576,3340,3342],{"className":3341},[684,685,686,687],[576,3343,774],{"className":3344},[650,687],[576,3346,852],{"className":3347},[851],[576,3349,3351],{"className":3350},[667],[576,3352,3354],{"className":3353,"style":1109},[671],[576,3355],{},[576,3357],{"className":3358,"style":1889},[697],[576,3360,1320],{"className":3361},[844],[576,3363],{"className":3364,"style":1889},[697],[576,3366,3368,3371,3374,3377,3380],{"className":3367},[641],[576,3369],{"className":3370,"style":1274},[645],[576,3372,1633],{"className":3373},[650],[576,3375],{"className":3376,"style":698},[697],[576,3378,609],{"className":3379},[702],[576,3381],{"className":3382,"style":698},[697],[576,3384,3386,3389],{"className":3385},[641],[576,3387],{"className":3388,"style":1274},[645],[576,3390,1242],{"className":3391},[650],[564,3393,3394,3395,2639,3608,3794,3795,1112],{},"Die positive Lösung ist ",[576,3396,3398,3433],{"className":3397},[579],[576,3399,3401],{"className":3400},[583],[585,3402,3403],{"xmlns":587},[589,3404,3405,3430],{},[592,3406,3407,3413,3415,3421,3423,3425,3427],{},[759,3408,3409,3411],{},[598,3410,600],{},[772,3412,774],{},[602,3414,609],{},[3416,3417,3418],"msqrt",{},[772,3419,3420],{},"3",[602,3422,1320],{},[772,3424,774],{},[602,3426,2281],{},[772,3428,3429],{},"0,7321",[630,3431,3432],{"encoding":632},"y_1 = \\sqrt{3} - 1 \\approx 0{,}7321",[576,3434,3436,3491,3571,3589],{"className":3435,"ariaHidden":622},[637],[576,3437,3439,3442,3482,3485,3488],{"className":3438},[641],[576,3440],{"className":3441,"style":1069},[645],[576,3443,3445,3448],{"className":3444},[650],[576,3446,600],{"className":3447,"style":655},[650,654],[576,3449,3451],{"className":3450},[659],[576,3452,3454,3474],{"className":3453},[663,818],[576,3455,3457,3471],{"className":3456},[667],[576,3458,3460],{"className":3459,"style":825},[671],[576,3461,3462,3465],{"style":888},[576,3463],{"className":3464,"style":680},[679],[576,3466,3468],{"className":3467},[684,685,686,687],[576,3469,774],{"className":3470},[650,687],[576,3472,852],{"className":3473},[851],[576,3475,3477],{"className":3476},[667],[576,3478,3480],{"className":3479,"style":1109},[671],[576,3481],{},[576,3483],{"className":3484,"style":698},[697],[576,3486,609],{"className":3487},[702],[576,3489],{"className":3490,"style":698},[697],[576,3492,3494,3498,3562,3565,3568],{"className":3493},[641],[576,3495],{"className":3496,"style":3497},[645],"height:1.04em;vertical-align:-0.1328em;",[576,3499,3502],{"className":3500},[650,3501],"sqrt",[576,3503,3505,3553],{"className":3504},[663,818],[576,3506,3508,3550],{"className":3507},[667],[576,3509,3512,3527],{"className":3510,"style":3511},[671],"height:0.9072em;",[576,3513,3517,3520],{"className":3514,"style":3516},[3515],"svg-align","top:-3em;",[576,3518],{"className":3519,"style":1429},[679],[576,3521,3524],{"className":3522,"style":3523},[650],"padding-left:0.833em;",[576,3525,3420],{"className":3526},[650],[576,3528,3530,3533],{"style":3529},"top:-2.8672em;",[576,3531],{"className":3532,"style":1429},[679],[576,3534,3538],{"className":3535,"style":3537},[3536],"hide-tail","min-width:0.853em;height:1.08em;",[3539,3540,3546],"svg",{"xmlns":3541,"width":3542,"height":3543,"viewBox":3544,"preserveAspectRatio":3545},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.08em","0 0 400000 1080","xMinYMin slice",[3547,3548],"path",{"d":3549},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[576,3551,852],{"className":3552},[851],[576,3554,3556],{"className":3555},[667],[576,3557,3560],{"className":3558,"style":3559},[671],"height:0.1328em;",[576,3561],{},[576,3563],{"className":3564,"style":1889},[697],[576,3566,1320],{"className":3567},[844],[576,3569],{"className":3570,"style":1889},[697],[576,3572,3574,3577,3580,3583,3586],{"className":3573},[641],[576,3575],{"className":3576,"style":1274},[645],[576,3578,774],{"className":3579},[650],[576,3581],{"className":3582,"style":698},[697],[576,3584,2281],{"className":3585},[702],[576,3587],{"className":3588,"style":698},[697],[576,3590,3592,3595,3598,3604],{"className":3591},[641],[576,3593],{"className":3594,"style":1695},[645],[576,3596,1242],{"className":3597},[650],[576,3599,3601],{"className":3600},[650],[576,3602,623],{"className":3603},[727],[576,3605,3607],{"className":3606},[650],"7321",[576,3609,3611,3644],{"className":3610},[579],[576,3612,3614],{"className":3613},[583],[585,3615,3616],{"xmlns":587},[589,3617,3618,3641],{},[592,3619,3620,3622,3624,3626,3628,3630,3636,3638],{},[598,3621,600],{},[602,3623,616],{"stretchy":615},[772,3625,2948],{},[602,3627,628],{"stretchy":615},[602,3629,609],{},[1306,3631,3632,3634],{},[772,3633,774],{},[772,3635,1928],{},[602,3637,2281],{},[772,3639,3640],{},"0,6667",[630,3642,3643],{"encoding":632},"y(0{,}5) = \\frac{1}{1{,}5} \\approx 0{,}6667",[576,3645,3647,3683,3775],{"className":3646,"ariaHidden":622},[637],[576,3648,3650,3653,3656,3659,3662,3668,3671,3674,3677,3680],{"className":3649},[641],[576,3651],{"className":3652,"style":712},[645],[576,3654,600],{"className":3655,"style":655},[650,654],[576,3657,616],{"className":3658},[720],[576,3660,1242],{"className":3661},[650],[576,3663,3665],{"className":3664},[650],[576,3666,623],{"className":3667},[727],[576,3669,1971],{"className":3670},[650],[576,3672,628],{"className":3673},[738],[576,3675],{"className":3676,"style":698},[697],[576,3678,609],{"className":3679},[702],[576,3681],{"className":3682,"style":698},[697],[576,3684,3686,3689,3766,3769,3772],{"className":3685},[641],[576,3687],{"className":3688,"style":2352},[645],[576,3690,3692,3695,3763],{"className":3691},[650],[576,3693],{"className":3694},[720,1409],[576,3696,3698],{"className":3697},[1306],[576,3699,3701,3755],{"className":3700},[663,818],[576,3702,3704,3752],{"className":3703},[667],[576,3705,3707,3730,3738],{"className":3706,"style":2023},[671],[576,3708,3709,3712],{"style":1425},[576,3710],{"className":3711,"style":1429},[679],[576,3713,3715],{"className":3714},[684,685,686,687],[576,3716,3718,3721,3727],{"className":3717},[650,687],[576,3719,774],{"className":3720},[650,687],[576,3722,3724],{"className":3723},[650,687],[576,3725,623],{"className":3726},[727,687],[576,3728,1971],{"className":3729},[650,687],[576,3731,3732,3735],{"style":1448},[576,3733],{"className":3734,"style":1429},[679],[576,3736],{"className":3737,"style":1456},[1455],[576,3739,3740,3743],{"style":2048},[576,3741],{"className":3742,"style":1429},[679],[576,3744,3746],{"className":3745},[684,685,686,687],[576,3747,3749],{"className":3748},[650,687],[576,3750,774],{"className":3751},[650,687],[576,3753,852],{"className":3754},[851],[576,3756,3758],{"className":3757},[667],[576,3759,3761],{"className":3760,"style":2425},[671],[576,3762],{},[576,3764],{"className":3765},[738,1409],[576,3767],{"className":3768,"style":698},[697],[576,3770,2281],{"className":3771},[702],[576,3773],{"className":3774,"style":698},[697],[576,3776,3778,3781,3784,3790],{"className":3777},[641],[576,3779],{"className":3780,"style":1695},[645],[576,3782,1242],{"className":3783},[650],[576,3785,3787],{"className":3786},[650],[576,3788,623],{"className":3789},[727],[576,3791,3793],{"className":3792},[650],"6667","; das explizite Verfahren liefert ",[576,3796,3798,3820],{"className":3797},[579],[576,3799,3801],{"className":3800},[583],[585,3802,3803],{"xmlns":587},[589,3804,3805,3817],{},[592,3806,3807,3809,3811,3813,3815],{},[772,3808,774],{},[602,3810,1320],{},[772,3812,2948],{},[602,3814,609],{},[772,3816,2948],{},[630,3818,3819],{"encoding":632},"1 - 0{,}5 = 0{,}5",[576,3821,3823,3841,3868],{"className":3822,"ariaHidden":622},[637],[576,3824,3826,3829,3832,3835,3838],{"className":3825},[641],[576,3827],{"className":3828,"style":1882},[645],[576,3830,774],{"className":3831},[650],[576,3833],{"className":3834,"style":1889},[697],[576,3836,1320],{"className":3837},[844],[576,3839],{"className":3840,"style":1889},[697],[576,3842,3844,3847,3850,3856,3859,3862,3865],{"className":3843},[641],[576,3845],{"className":3846,"style":1695},[645],[576,3848,1242],{"className":3849},[650],[576,3851,3853],{"className":3852},[650],[576,3854,623],{"className":3855},[727],[576,3857,1971],{"className":3858},[650],[576,3860],{"className":3861,"style":698},[697],[576,3863,609],{"className":3864},[702],[576,3866],{"className":3867,"style":698},[697],[576,3869,3871,3874,3877,3883],{"className":3870},[641],[576,3872],{"className":3873,"style":1695},[645],[576,3875,1242],{"className":3876},[650],[576,3878,3880],{"className":3879},[650],[576,3881,623],{"className":3882},[727],[576,3884,1971],{"className":3885},[650],[559,3887,3889],{"id":3888},"typische-fehler","Typische Fehler",[1117,3891,3892,3895,3967,3970],{},[1120,3893,3894],{},"Die Steigung am alten Punkt auswerten. Das ist das explizite Euler-Verfahren.",[1120,3896,3897,3898,1112],{},"Bei nichtlinearen Gleichungen die falsche Lösung der Schrittgleichung wählen. Gesucht ist die Lösung nahe ",[576,3899,3901,3918],{"className":3900},[579],[576,3902,3904],{"className":3903},[583],[585,3905,3906],{"xmlns":587},[589,3907,3908,3916],{},[592,3909,3910],{},[759,3911,3912,3914],{},[598,3913,600],{},[598,3915,767],{},[630,3917,1059],{"encoding":632},[576,3919,3921],{"className":3920,"ariaHidden":622},[637],[576,3922,3924,3927],{"className":3923},[641],[576,3925],{"className":3926,"style":1069},[645],[576,3928,3930,3933],{"className":3929},[650],[576,3931,600],{"className":3932,"style":655},[650,654],[576,3934,3936],{"className":3935},[659],[576,3937,3939,3959],{"className":3938},[663,818],[576,3940,3942,3956],{"className":3941},[667],[576,3943,3945],{"className":3944,"style":1088},[671],[576,3946,3947,3950],{"style":888},[576,3948],{"className":3949,"style":680},[679],[576,3951,3953],{"className":3952},[684,685,686,687],[576,3954,767],{"className":3955},[650,654,687],[576,3957,852],{"className":3958},[851],[576,3960,3962],{"className":3961},[667],[576,3963,3965],{"className":3964,"style":1109},[671],[576,3966],{},[1120,3968,3969],{},"Stabilität mit Genauigkeit verwechseln: Auch das implizite Verfahren hat nur Ordnung 1. Große Schritte bleiben stabil, sind aber ungenau.",[1120,3971,3972,3973,4002,4003,4031,4032,4103,4104,4190],{},"Bei ",[576,3974,3976,3989],{"className":3975},[579],[576,3977,3979],{"className":3978},[583],[585,3980,3981],{"xmlns":587},[589,3982,3983,3987],{},[592,3984,3985],{},[598,3986,619],{},[630,3988,619],{"encoding":632},[576,3990,3992],{"className":3991,"ariaHidden":622},[637],[576,3993,3995,3999],{"className":3994},[641],[576,3996],{"className":3997,"style":3998},[645],"height:0.6151em;",[576,4000,619],{"className":4001},[650,654],"-abhängigem ",[576,4004,4006,4019],{"className":4005},[579],[576,4007,4009],{"className":4008},[583],[585,4010,4011],{"xmlns":587},[589,4012,4013,4017],{},[592,4014,4015],{},[598,4016,612],{},[630,4018,612],{"encoding":632},[576,4020,4022],{"className":4021,"ariaHidden":622},[637],[576,4023,4025,4028],{"className":4024},[641],[576,4026],{"className":4027,"style":1034},[645],[576,4029,612],{"className":4030,"style":716},[650,654]," den alten Zeitpunkt ",[576,4033,4035,4053],{"className":4034},[579],[576,4036,4038],{"className":4037},[583],[585,4039,4040],{"xmlns":587},[589,4041,4042,4050],{},[592,4043,4044],{},[759,4045,4046,4048],{},[598,4047,619],{},[598,4049,767],{},[630,4051,4052],{"encoding":632},"t_n",[576,4054,4056],{"className":4055,"ariaHidden":622},[637],[576,4057,4059,4063],{"className":4058},[641],[576,4060],{"className":4061,"style":4062},[645],"height:0.7651em;vertical-align:-0.15em;",[576,4064,4066,4069],{"className":4065},[650],[576,4067,619],{"className":4068},[650,654],[576,4070,4072],{"className":4071},[659],[576,4073,4075,4095],{"className":4074},[663,818],[576,4076,4078,4092],{"className":4077},[667],[576,4079,4081],{"className":4080,"style":1088},[671],[576,4082,4083,4086],{"style":828},[576,4084],{"className":4085,"style":680},[679],[576,4087,4089],{"className":4088},[684,685,686,687],[576,4090,767],{"className":4091},[650,654,687],[576,4093,852],{"className":4094},[851],[576,4096,4098],{"className":4097},[667],[576,4099,4101],{"className":4100,"style":1109},[671],[576,4102],{}," statt ",[576,4105,4107,4131],{"className":4106},[579],[576,4108,4110],{"className":4109},[583],[585,4111,4112],{"xmlns":587},[589,4113,4114,4128],{},[592,4115,4116],{},[759,4117,4118,4120],{},[598,4119,619],{},[592,4121,4122,4124,4126],{},[598,4123,767],{},[602,4125,770],{},[772,4127,774],{},[630,4129,4130],{"encoding":632},"t_{n+1}",[576,4132,4134],{"className":4133,"ariaHidden":622},[637],[576,4135,4137,4141],{"className":4136},[641],[576,4138],{"className":4139,"style":4140},[645],"height:0.8234em;vertical-align:-0.2083em;",[576,4142,4144,4147],{"className":4143},[650],[576,4145,619],{"className":4146},[650,654],[576,4148,4150],{"className":4149},[659],[576,4151,4153,4182],{"className":4152},[663,818],[576,4154,4156,4179],{"className":4155},[667],[576,4157,4159],{"className":4158,"style":825},[671],[576,4160,4161,4164],{"style":828},[576,4162],{"className":4163,"style":680},[679],[576,4165,4167],{"className":4166},[684,685,686,687],[576,4168,4170,4173,4176],{"className":4169},[650,687],[576,4171,767],{"className":4172},[650,654,687],[576,4174,770],{"className":4175},[844,687],[576,4177,774],{"className":4178},[650,687],[576,4180,852],{"className":4181},[851],[576,4183,4185],{"className":4184},[667],[576,4186,4188],{"className":4187,"style":859},[671],[576,4189],{}," einsetzen.",{"title":4192,"searchDepth":4193,"depth":4193,"links":4194},"",2,[4195,4196,4197],{"id":561,"depth":4193,"text":562},{"id":1605,"depth":4193,"text":1606},{"id":3888,"depth":4193,"text":3889},[4199,4204,4207,4210,4213,4216,4219,4221,4224,4227],{"lhs":4200,"rhs":4201,"where":4202},"y\u002F(1 - h*l)","y + h*l*y\u002F(1 - h*l)",{"l":4203},-2,{"lhs":4205,"rhs":4206},"1\u002F(1 + 1.5*2)","0.25",{"lhs":4208,"rhs":4209},"0.25^4","1\u002F256",{"lhs":4211,"rhs":4212},"round(1\u002F(1 + 0.1*2), 4)","0.8333",{"lhs":4214,"rhs":4215},"round(1.2^(-10), 4)","0.1615",{"lhs":4217,"rhs":4218},"round(exp(-2), 4)","0.1353",{"lhs":4220,"rhs":1242},"0.5*(sqrt(3) - 1)^2 + (sqrt(3) - 1) - 1",{"lhs":4222,"rhs":4223},"round(sqrt(3) - 1, 4)","0.7321",{"lhs":4225,"rhs":4226},"round(1\u002F(1 + 0.5), 4)","0.6667",{"lhs":4228,"rhs":4229},"1 - 0.5*1^2","0.5","y' = f(t, y),\\ y(t_0) = y_0,\\ h > 0 \\text{ (Schrittgleichung eindeutig lösbar, z.B. bei } hL \u003C 1 \\text{ mit Lipschitz-Konstante } L\\text{)}","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmi>y\u003C\u002Fmi>\u003Cmo mathvariant=\"normal\" lspace=\"0em\" rspace=\"0em\">′\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>f\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>t\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmi>y\u003C\u002Fmi>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">y' = f(t, y),\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.9463em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7519em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">′\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">t\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>y\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>t\u003C\u002Fmi>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmsub>\u003Cmi>y\u003C\u002Fmi>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">y(t_0) = y_0,\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">t\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>h\u003C\u002Fmi>\u003Cmo>>\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">h > 0\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.7335em;vertical-align:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">h\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">(Schrittgleichung eindeutig lösbar, z.B. bei\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>h\u003C\u002Fmi>\u003Cmi>L\u003C\u002Fmi>\u003Cmo><\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">hL < 1\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.7335em;vertical-align:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">h\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\"><\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">mit Lipschitz-Konstante\u003C\u002Fspan> \u003Cspan style=\"white-space: nowrap\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>L\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">L\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">L\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex\">)\u003C\u002Fspan>\u003C\u002Fspan>",null,"md","implicit-euler","y_{n+1} = y_n + h \\cdot f(t_{n+1}, y_{n+1}), \\qquad t_{n+1} = t_n + h","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>y\u003C\u002Fmi>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>=\u003C\u002Fmo>\u003Cmsub>\u003Cmi>y\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>+\u003C\u002Fmo>\u003Cmi>h\u003C\u002Fmi>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmi>f\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>t\u003C\u002Fmi>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>y\u003C\u002Fmi>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmspace width=\"2em\"\u002F>\u003Cmsub>\u003Cmi>t\u003C\u002Fmi>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003Cmo>=\u003C\u002Fmo>\u003Cmsub>\u003Cmi>t\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>+\u003C\u002Fmo>\u003Cmi>h\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">y_{n+1} = y_n + h \\cdot f(t_{n+1}, y_{n+1}), \\qquad t_{n+1} = t_n + h\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6389em;vertical-align:-0.2083em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t 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style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">h\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">⋅\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.1076em;\">f\u003C\u002Fspan>\u003Cspan 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style=\"height:0.2083em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:2em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">t\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003Cspan class=\"mbin mtight\">+\u003C\u002Fspan>\u003Cspan class=\"mord 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style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord 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xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>y\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">y_n\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0359em;\">y\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan 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