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Rainbow Solution Curves of ODEs with Slope Field

Rainbow Solution Curves of ODEs with Slope Field
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2019-09-08 11:48
FunctionCalculusParametric EquationSlider AnimationGraphing
① This GeoGebra example demonstrates a family of rainbow-colored solution curves for ordinary differential equations (ODEs) overlaid on a slope field, showing how solutions evolve from various initial points. ② Core mathematical concepts include first-order ODEs, slope fields (direction fields), initial value problems, uniqueness and branching of solutions, and parameter dependence. ③ Construction highlights: the SolveODE command computes solution curves, SlopeField generates the direction field; sliders n and speed control the number of curves and animation speed; a button triggers dynamic updates; points A, Ax, Ay define initial conditions; sol stores the solution; A1–A18 mark distinct initial points; parameters a and b adjust the equation; slp represents slope. ④ This is ideal for university-level calculus or differential equations courses, and for advanced high school students, to visually explore how initial conditions and parameters shape solution families and to connect analytic solutions with geometric slope fields.

GeoGebra Commands & Parameters

  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