一阶常微分方程 RK4 数值解
已被使用 — 次
支持 sin cos tan exp log sqrt pow abs 与常量 pi、e，变量 x、y；乘法需显式写*，幂用^或pow。
常用方程示例库（点击一键填充）：
解曲线 y(x)
数值采样表（均匀间隔，仅展示部分节点）
| x | y(x) |
|---|---|
| 0 | 1 |
| 0.3 | 0.74918215 |
| 0.5 | 0.64346993 |
| 0.8 | 0.59067192 |
| 1.1 | 0.67713005 |
| 1.3 | 0.81746948 |
| 1.6 | 1.1581049 |
| 1.8 | 1.4747027 |
| 2.1 | 2.0875453 |
| 2.4 | 2.8692838 |
| 2.6 | 3.4857283 |
| 2.9 | 4.5549787 |
| 3.2 | 5.7992398 |
| 3.4 | 6.7266287 |
| 3.7 | 8.2652785 |
| 3.9 | 9.3897602 |
| 4.2 | 11.225007 |
| 4.5 | 13.238893 |
| 4.7 | 14.680907 |
| 5 | 16.993264 |
完整 51 个节点见下方 JSON。
求解数据（JSON）
{
"solver": "RK4 (四阶龙格-库塔)",
"equation": "x*x - y",
"initial": {
"x0": 0,
"y0": 1
},
"step_size": 0.1,
"points_count": 51,
"sampled_data": [
{
"x": 0,
"y": 1
},
{
"x": 0.1,
"y": 0.9051627083333333
},
{
"x": 0.2,
"y": 0.8212694954348958
},
{
"x": 0.30000000000000004,
"y": 0.7491821454089058
},
{
"x": 0.4,
"y": 0.6896804328297642
},
{
"x": 0.5,
"y": 0.6434699269739351
},
{
"x": 0.6,
"y": 0.6111890533816113
},
{
"x": 0.7,
"y": 0.593415483422517
},
{
"x": 0.7999999999999999,
"y": 0.5906719158146551
},
{
"x": 0.8999999999999999,
"y": 0.6034313079592762
},
{
"x": 0.9999999999999999,
"y": 0.6321216094489349
},
{
"x": 1.0999999999999999,
"y": 0.677130045123084
},
{
"x": 1.2,
"y": 0.7388069905373917
},
{
"x": 1.3,
"y": 0.8174694786337106
},
{
"x": 1.4000000000000001,
"y": 0.9134043727065634
},
{
"x": 1.5000000000000002,
"y": 1.0268712374222084
},
{
"x": 1.6000000000000003,
"y": 1.158104936624351
},
{
"x": 1.7000000000000004,
"y": 1.3073179839261697
},
{
"x": 1.8000000000000005,
"y": 1.474702669614129
},
{
"x": 1.9000000000000006,
"y": 1.660432985150308
},
{
"x": 2.0000000000000004,
"y": 1.8646663645342754
},
{
"x": 2.1000000000000005,
"y": 2.087545259952616
},
{
"x": 2.2000000000000006,
"y": 2.3291985674857085
},
{
"x": 2.3000000000000007,
"y": 2.589742917140683
},
{
"x": 2.400000000000001,
"y": 2.8692838401216165
},
{
"x": 2.500000000000001,
"y": 3.1679168250193768
},
{
"x": 2.600000000000001,
"y": 3.4857282734918043
},
{
"x": 2.700000000000001,
"y": 3.822796364998974
},
{
"x": 2.800000000000001,
"y": 4.179191839248093
},
{
"x": 2.9000000000000012,
"y": 4.55497870417898
},
{
"x": 3.0000000000000013,
"y": 4.950214876575882
},
{
"x": 3.1000000000000014,
"y": 5.364952761717063
},
{
"x": 3.2000000000000015,
"y": 5.799239777863497
},
{
"x": 3.3000000000000016,
"y": 6.253118830835897
},
{
"x": 3.4000000000000017,
"y": 6.72662874342981
},
{
"x": 3.5000000000000018,
"y": 7.219804643966505
},
{
"x": 3.600000000000002,
"y": 7.732678317868377
},
{
"x": 3.700000000000002,
"y": 8.265278525777562
},
{
"x": 3.800000000000002,
"y": 8.81763129140159
},
{
"x": 3.900000000000002,
"y": 9.38976016196692
},
{
"x": 4.000000000000002,
"y": 9.981686443887078
},
{
"x": 4.100000000000001,
"y": 10.593429416004009
},
{
"x": 4.200000000000001,
"y": 11.225006522536862
},
{
"x": 4.300000000000001,
"y": 11.876433547669283
},
{
"x": 4.4,
"y": 12.547724773522539
},
{
"x": 4.5,
"y": 13.238893123095533
},
{
"x": 4.6,
"y": 13.949950289602286
},
{
"x": 4.699999999999999,
"y": 14.680906853501341
},
{
"x": 4.799999999999999,
"y": 15.431772388388351
},
{
"x": 4.899999999999999,
"y": 16.202555556811678
},
{
"x": 4.999999999999998,
"y": 16.993264196969918
}
],
"final_point": {
"x": 4.999999999999998,
"y": 16.993264196969918
}
}所有计算均在浏览器本地完成，不会上传任何数据。
下表为采用四阶龙格-库塔法（RK4）在区间 [0, 5]、步长 0.1下计算的数值解（均匀采样 20 个节点）。完整数据可在上方工具中点击「复制 JSON」获取。
| x | y(x) |
|---|---|
| 0 | 1 |
| 0.3 | 0.749182 |
| 0.5 | 0.64347 |
| 0.8 | 0.590672 |
| 1.1 | 0.67713 |
| 1.3 | 0.817469 |
| 1.6 | 1.158105 |
| 1.8 | 1.474703 |
| 2.1 | 2.087545 |
| 2.4 | 2.869284 |
| 2.6 | 3.485728 |
| 2.9 | 4.554979 |
| 3.2 | 5.79924 |
| 3.4 | 6.726629 |
| 3.7 | 8.265279 |
| 3.9 | 9.38976 |
| 4.2 | 11.225007 |
| 4.5 | 13.238893 |
| 4.7 | 14.680907 |
| 5 | 16.993264 |