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MokaKit

逻辑斯蒂增长：解 y' = y*(1 - y)

一阶常微分方程 RK4 数值解

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）：

共计算 101 个点终点 y(10) = 0.99959157y 范围：0.1 ~ 0.99959157

解曲线 y(x)

0100.9996

数值采样表（均匀间隔，仅展示部分节点）

xy(x)
00.1
0.50.15482807
1.10.25026022
1.60.35497882
2.10.47571287
2.60.59935418
3.20.7316038
3.70.81798795
4.20.88108808
4.70.92433608
5.30.95700702
5.80.97347472
6.30.98374191
6.80.99007548
7.40.99452881
7.90.99667439
8.40.99798027
8.90.998774
9.50.99932678
100.99959157

完整 101 个节点见下方 JSON。

求解数据（JSON）

展开查看
{
  "solver": "RK4 (四阶龙格-库塔)",
  "equation": "y*(1 - y)",
  "initial": {
    "x0": 0,
    "y0": 0.1
  },
  "step_size": 0.1,
  "points_count": 101,
  "sampled_data": [
    {
      "x": 0,
      "y": 0.1
    },
    {
      "x": 0.1,
      "y": 0.10936686601837953
    },
    {
      "x": 0.2,
      "y": 0.11949462250734001
    },
    {
      "x": 0.30000000000000004,
      "y": 0.130422905547995
    },
    {
      "x": 0.4,
      "y": 0.1421892310344913
    },
    {
      "x": 0.5,
      "y": 0.15482807269260507
    },
    {
      "x": 0.6,
      "y": 0.1683698432971501
    },
    {
      "x": 0.7,
      "y": 0.18283979375391615
    },
    {
      "x": 0.7999999999999999,
      "y": 0.1982568521779063
    },
    {
      "x": 0.8999999999999999,
      "y": 0.21463243321907088
    },
    {
      "x": 0.9999999999999999,
      "y": 0.23196925620916956
    },
    {
      "x": 1.0999999999999999,
      "y": 0.2502602186109049
    },
    {
      "x": 1.2,
      "y": 0.2694873779735329
    },
    {
      "x": 1.3,
      "y": 0.28962110026076804
    },
    {
      "x": 1.4000000000000001,
      "y": 0.3106194341083855
    },
    {
      "x": 1.5000000000000002,
      "y": 0.33242776845664024
    },
    {
      "x": 1.6000000000000003,
      "y": 0.35497882445601087
    },
    {
      "x": 1.7000000000000004,
      "y": 0.3781930212719432
    },
    {
      "x": 1.8000000000000005,
      "y": 0.4019792395811285
    },
    {
      "x": 1.9000000000000006,
      "y": 0.4262359868591392
    },
    {
      "x": 2.0000000000000004,
      "y": 0.4508529462524189
    },
    {
      "x": 2.1000000000000005,
      "y": 0.47571286762274867
    },
    {
      "x": 2.2000000000000006,
      "y": 0.5006937372748621
    },
    {
      "x": 2.3000000000000007,
      "y": 0.5256711440295317
    },
    {
      "x": 2.400000000000001,
      "y": 0.5505207455904755
    },
    {
      "x": 2.500000000000001,
      "y": 0.5751207320161708
    },
    {
      "x": 2.600000000000001,
      "y": 0.5993541833149003
    },
    {
      "x": 2.700000000000001,
      "y": 0.6231112257106346
    },
    {
      "x": 2.800000000000001,
      "y": 0.6462909051694145
    },
    {
      "x": 2.9000000000000012,
      "y": 0.6688027158564376
    },
    {
      "x": 3.0000000000000013,
      "y": 0.6905677433896442
    },
    {
      "x": 3.1000000000000014,
      "y": 0.7115194059618222
    },
    {
      "x": 3.2000000000000015,
      "y": 0.7316037986062317
    },
    {
      "x": 3.3000000000000016,
      "y": 0.7507796653870891
    },
    {
      "x": 3.4000000000000017,
      "y": 0.7690180398895771
    },
    {
      "x": 3.5000000000000018,
      "y": 0.7863016053942946
    },
    {
      "x": 3.600000000000002,
      "y": 0.8026238324102307
    },
    {
      "x": 3.700000000000002,
      "y": 0.8179879531225195
    },
    {
      "x": 3.800000000000002,
      "y": 0.832405830431725
    },
    {
      "x": 3.900000000000002,
      "y": 0.8458967744614309
    },
    {
      "x": 4.000000000000002,
      "y": 0.8584863525998238
    },
    {
      "x": 4.100000000000001,
      "y": 0.8702052311914944
    },
    {
      "x": 4.200000000000001,
      "y": 0.881088078669822
    },
    {
      "x": 4.300000000000001,
      "y": 0.8911725518268756
    },
    {
      "x": 4.4,
      "y": 0.9004983794978354
    },
    {
      "x": 4.5,
      "y": 0.909106551470644
    },
    {
      "x": 4.6,
      "y": 0.9170386150590353
    },
    {
      "x": 4.699999999999999,
      "y": 0.9243360775264426
    },
    {
      "x": 4.799999999999999,
      "y": 0.9310399093659721
    },
    {
      "x": 4.899999999999999,
      "y": 0.9371901412203757
    },
    {
      "x": 4.999999999999998,
      "y": 0.9428255458288767
    },
    {
      "x": 5.099999999999998,
      "y": 0.947983395667044
    },
    {
      "x": 5.1999999999999975,
      "y": 0.9526992867567575
    },
    {
      "x": 5.299999999999997,
      "y": 0.9570070193324925
    },
    {
      "x": 5.399999999999997,
      "y": 0.9609385265413358
    },
    {
      "x": 5.4999999999999964,
      "y": 0.9645238430299786
    },
    {
      "x": 5.599999999999996,
      "y": 0.9677911060540695
    },
    {
      "x": 5.699999999999996,
      "y": 0.9707665825731158
    },
    {
      "x": 5.799999999999995,
      "y": 0.9734747166225716
    },
    {
      "x": 5.899999999999995,
      "y": 0.9759381920520481
    },
    {
      "x": 5.999999999999995,
      "y": 0.9781780064636715
    },
    {
      "x": 6.099999999999994,
      "y": 0.9802135528649246
    },
    {
      "x": 6.199999999999994,
      "y": 0.9820627061597116
    },
    {
      "x": 6.299999999999994,
      "y": 0.983741912138447
    },
    {
      "x": 6.399999999999993,
      "y": 0.9852662770945954
    },
    {
      "x": 6.499999999999993,
      "y": 0.9866496565953253
    },
    {
      "x": 6.5999999999999925,
      "y": 0.9879047422731835
    },
    {
      "x": 6.699999999999992,
      "y": 0.9890431457899989
    },
    {
      "x": 6.799999999999992,
      "y": 0.9900754793598541
    },
    {
      "x": 6.8999999999999915,
      "y": 0.9910114324110584
    },
    {
      "x": 6.999999999999991,
      "y": 0.9918598441234574
    },
    {
      "x": 7.099999999999991,
      "y": 0.9926287717024721
    },
    {
      "x": 7.19999999999999,
      "y": 0.993325554349844
    },
    {
      "x": 7.29999999999999,
      "y": 0.9939568729674649
    },
    {
      "x": 7.39999999999999,
      "y": 0.9945288056886802
    },
    {
      "x": 7.499999999999989,
      "y": 0.9950468793743354
    },
    {
      "x": 7.599999999999989,
      "y": 0.9955161172413924
    },
    {
      "x": 7.699999999999989,
      "y": 0.9959410828125558
    },
    {
      "x": 7.799999999999988,
      "y": 0.9963259203880246
    },
    {
      "x": 7.899999999999988,
      "y": 0.9966743922468991
    },
    {
      "x": 7.999999999999988,
      "y": 0.9969899127873075
    },
    {
      "x": 8.099999999999987,
      "y": 0.9972755798121306
    },
    {
      "x": 8.199999999999987,
      "y": 0.9975342031622143
    },
    {
      "x": 8.299999999999986,
      "y": 0.9977683308919416
    },
    {
      "x": 8.399999999999986,
      "y": 0.9979802731735823
    },
    {
      "x": 8.499999999999986,
      "y": 0.9981721241074556
    },
    {
      "x": 8.599999999999985,
      "y": 0.9983457816049932
    },
    {
      "x": 8.699999999999985,
      "y": 0.9985029655015876
    },
    {
      "x": 8.799999999999985,
      "y": 0.9986452340458831
    },
    {
      "x": 8.899999999999984,
      "y": 0.9987739989020871
    },
    {
      "x": 8.999999999999984,
      "y": 0.9988905387920707
    },
    {
      "x": 9.099999999999984,
      "y": 0.9989960118945979
    },
    {
      "x": 9.199999999999983,
      "y": 0.9990914671100141
    },
    {
      "x": 9.299999999999983,
      "y": 0.9991778542901979
    },
    {
      "x": 9.399999999999983,
      "y": 0.9992560335255433
    },
    {
      "x": 9.499999999999982,
      "y": 0.9993267835732084
    },
    {
      "x": 9.599999999999982,
      "y": 0.9993908095038401
    },
    {
      "x": 9.699999999999982,
      "y": 0.999448749637443
    },
    {
      "x": 9.799999999999981,
      "y": 0.9995011818330068
    },
    {
      "x": 9.89999999999998,
      "y": 0.9995486291908993
    },
    {
      "x": 9.99999999999998,
      "y": 0.9995915652218679
    }
  ],
  "final_point": {
    "x": 9.99999999999998,
    "y": 0.9995915652218679
  }
}

所有计算均在浏览器本地完成，不会上传任何数据。

逻辑斯蒂增长：解 y' = y*(1 - y) 数值解采样表

下表为采用四阶龙格-库塔法（RK4）在区间 [0, 10]、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。

xy(x)
00.1
0.50.154828
1.10.25026
1.60.354979
2.10.475713
2.60.599354
3.20.731604
3.70.817988
4.20.881088
4.70.924336
5.30.957007
5.80.973475
6.30.983742
6.80.990075
7.40.994529
7.90.996674
8.40.99798
8.90.998774
9.50.999327
100.999592

常见问题

什么是微分方程初值问题？
给定一个方程 dy/dx = f(x, y) 和起点 (x₀, y₀)，求 y 随 x 变化的函数。初值条件 y(x₀)=y₀ 唯一确定了数值解的路径。
RK4 是什么？精度如何？
四阶龙格-库塔法（Runge-Kutta），局部截断误差为 O(h⁵)，是工程与科研中最常用的高精度数值积分方法，比欧拉法稳定且精确得多。
步长 h 应该怎么选？
步长越小精度越高，但采样点越多、计算越慢。推荐 h 在 0.01~0.1 之间；若步数超过 2000 本工具会提示优化。解变化剧烈时可适当减小 h。
为什么很多方程没有解析解？
只有少数特殊形式（可分离变量、线性、恰当方程等）能写出闭式解。多数非线性方程没有解析解，数值解（如 RK4）是唯一可行的途径，也能画出解曲线直观观察。

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