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常微分方程的舞动解曲线

常微分方程的舞动解曲线
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2019-09-08 11:50
函数微积分最值优化参数方程滑块动画
①本范例演示了常微分方程的解曲线随参数变化而产生的动态舞动效果；②涉及的核心数学概念包括常微分方程、方向场、解曲线、参数化动画；③作图要点——从下方『构造摘要』提炼用到了 SolveODE, SlopeField, t, f, button1, Ax, Ay, A, sol, n, speed, slp, A1, A2, A3, A4, A5, A6, A7, A8, A9, A10, A11, A12, A13, A14, A15, A16, A17, A18, a, B1, B2, B3, B4, B5, B6, B7, B8, B9, B10, B11 等GeoGebra命令/对象；④适合大学数学、物理专业学生学习常微分方程的数值解与可视化表现，也可用于教学演示。

GeoGebra 命令及参数

  1. SolveODE(f(x, y), Ax, Ay, 100, 0.08) → sol
  2. SlopeField(f) → slp
  3. SolveODE(f(x, y), Ax, Ay + 0 - 0.8, 100, 0.08) → A1
  4. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (2 - 1)) - 0.8, 100, 0.08) → A2
  5. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (3 - 1)) - 0.8, 100, 0.08) → A3
  6. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (4 - 1)) - 0.8, 100, 0.08) → A4
  7. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (5 - 1)) - 0.8, 100, 0.08) → A5
  8. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (6 - 1)) - 0.8, 100, 0.08) → A6
  9. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (7 - 1)) - 0.8, 100, 0.08) → A7
  10. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (8 - 1)) - 0.8, 100, 0.08) → A8
  11. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (9 - 1)) - 0.8, 100, 0.08) → A9
  12. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (10 - 1)) - 0.8, 100, 0.08) → A10
  13. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (11 - 1)) - 0.8, 100, 0.08) → A11
  14. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (12 - 1)) - 0.8, 100, 0.08) → A12
  15. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (13 - 1)) - 0.8, 100, 0.08) → A13
  16. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (14 - 1)) - 0.8, 100, 0.08) → A14
  17. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (15 - 1)) - 0.8, 100, 0.08) → A15
  18. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (16 - 1)) - 0.8, 100, 0.08) → A16
  19. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (17 - 1)) - 0.8, 100, 0.08) → A17
  20. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (18 - 1)) - 0.8, 100, 0.08) → A18
  21. SolveODE(f(x, y), Ax, Ay + 0 - 0.8, -100, 0.08) → B1
  22. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (2 - 1)) - 0.8, -100, 0.08) → B2
  23. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (3 - 1)) - 0.8, -100, 0.08) → B3
  24. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (4 - 1)) - 0.8, -100, 0.08) → B4
  25. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (5 - 1)) - 0.8, -100, 0.08) → B5
  26. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (6 - 1)) - 0.8, -100, 0.08) → B6
  27. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (7 - 1)) - 0.8, -100, 0.08) → B7
  28. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (8 - 1)) - 0.8, -100, 0.08) → B8
  29. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (9 - 1)) - 0.8, -100, 0.08) → B9
  30. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (10 - 1)) - 0.8, -100, 0.08) → B10
  31. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (11 - 1)) - 0.8, -100, 0.08) → B11
  32. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (12 - 1)) - 0.8, -100, 0.08) → B12
  33. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (13 - 1)) - 0.8, -100, 0.08) → B13
  34. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (14 - 1)) - 0.8, -100, 0.08) → B14
  35. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (15 - 1)) - 0.8, -100, 0.08) → B15
  36. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (16 - 1)) - 0.8, -100, 0.08) → B16
  37. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (17 - 1)) - 0.8, -100, 0.08) → B17
  38. SolveODE(f(x, y), Ax, Ay + (13 / 170 * (18 - 1)) - 0.8, -100, 0.08) → B18

AI 绘图提示语参考

绘制一组常微分方程的解曲线，展示它们随参数变化的动态舞动效果。首先绘制基础解曲线，然后基于该曲线生成多个解曲线，通过调整斜率参数使它们围绕基础曲线动态变化，形成舞动效果。同时显示斜率场作为背景，帮助理解解曲线的动态变化规律。
以上提示语为根据原始绘图命令整理而来，仅供参考。