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一阶线性齐次微分方程的相图

一阶线性齐次微分方程的相图
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2016-10-13 14:51
向量函数微积分函数图像
所属专题 UQ-Applied Mathematical Analysis
①本范例演示了一阶线性齐次微分方程的相图；②涉及的核心数学概念包括微分方程、线性方程、对数函数、相似三角形、频率分布、直角三角形；③作图要点提炼自构造摘要，使用了Identity, Element, If, Textfield, UnitVector, Line, SurdText, Locus, Sequence, Point, Segment, Mirror, Determinant, LaTeX等GeoGebra命令/对象；④适合大学数学专业学生或对微分方程图形化理解有需求的学习者。

GeoGebra 命令及参数

  1. Identity(2) → I
  2. Element(eigenvalues, 2) → λ2
  3. Element(eigenvalues, 1) → λ1
  4. If(c ≠ 0 ∨ a - λ1 ≠ 0, Vector[((sgn(a - λ1) * (a - λ1)), (sgn(a - λ1) * c))], Vector[If[b ≠ 0 ∨ d - λ1 ≠ 0, Vector[((sgn(b) * b), (sgn(b) * (d - λ1)))], Vector[(1, 0)]]]) → Ev2
  5. If(c ≟ 0 ∧ b ≟ 0 ∧ a > d, Vector[(1, 0)], Vector[If[c ≟ 0 ∧ b ≟ 0 ∧ a ≤ d, Vector[(1, 0)]]]) → w2
  6. If(c ≟ 0 ∧ b ≟ 0 ∧ a > d, Vector[(0, 1)], Vector[If[c ≟ 0 ∧ b ≟ 0 ∧ a ≤ d, Vector[(0, 1)]]]) → w1
  7. Textfield(a) → InputBox1
  8. Textfield(b) → InputBox2
  9. Textfield(c) → InputBox3
  10. Textfield(d) → InputBox4
  11. If(a^(2) + ((4 * b) * c) - ((2 * a) * d) + d^(2) > 0, 2, If[a^(2) + ((4 * b) * c) - ((2 * a) * d) + d^(2) < 0, 0, 1]) → numEigenvalues
  12. UnitVector(Vector[If[eive2 ≟ eive2, eive2, Vector[If[w2 ≟ w2, w2, Ev2]]]]) → e2
  13. UnitVector(Vector[If[eive1 ≟ eive1, eive1, Vector[If[numEigenvalues ≟ 1, Ev2, Vector[If[w1 ≟ w1, w1, Ev1]]]]]]) → e1
  14. Line((0, 0), e1) → h
  15. Line((0, 0), e2) → i
  16. SurdText(x(e1)) → text8
  17. SurdText(y(e1)) → text9
  18. SurdText(x(e2)) → text10
  19. SurdText(y(e2)) → text11
  20. SurdText(λ1) → text14
  21. SurdText(λ2) → text15
  22. Locus(g(x, y) / f(x, y), A) → numericalIntegral1
  23. Sequence((x(Element[ListP, i]) + (0.001 * f(x(Element[ListP, i]), y(Element[ListP, i]))), y(Element[ListP, i]) + (0.001 * g(x(Element[ListP, i]), y(Element[ListP, i])))), i, 1, 8) → ListP1
  24. Sequence(Segment[Element[ListP, i], Element[ListP1, i]], i, 1, 8) → SegmentList
  25. Element(SegmentList, 1) → Segment1
  26. Element(SegmentList, 2) → Segment2
  27. Element(SegmentList, 3) → Segment3
  28. Element(SegmentList, 4) → Segment4
  29. Element(SegmentList, 5) → Segment5
  30. Element(SegmentList, 6) → Segment6
  31. Element(SegmentList, 7) → Segment7
  32. Sequence(Locus[g(x, y) / f(x, y), Element[ListP, i]], i, 1, 8) → list1
  33. Element(SegmentList, 8) → Segment8
  34. Point(h) → N
  35. Point(i) → O
  36. Segment(N, N1) → Segment9
  37. Segment(O, O1) → Segment10
  38. Mirror(N, (0, 0)) → N'
  39. Mirror(N1, (0, 0)) → N1'
  40. Segment(N', N1') → Segment9'
  41. Mirror(O, (0, 0)) → O'
  42. Mirror(O1, (0, 0)) → O1'
  43. Segment(O', O1') → Segment10'
  44. Determinant(T) → Determinant
  45. Textfield(a) → InputBox1_1
  46. Textfield(b) → InputBox2_1
  47. Textfield(c) → InputBox3_1
  48. Textfield(d) → InputBox4_1
  49. Determinant(T_1) → Determinant_1
  50. If(eive2 ≟ eive2, eive2, Vector[If[w2 ≟ w2, w2, Ev2]]) → e22
  51. If(eive1 ≟ eive1, eive1, Vector[If[numEigenvalues ≟ 1, Ev2, Vector[If[w1 ≟ w1, w1, Ev1]]]]) → e11
  52. Element(CEigenvalues, 1) → cλ1
  53. Element(CEigenvalues, 2) → cλ2
  54. Identity(2) → I_1
  55. Element(eigenvalues_1, 1) → λ1_1
  56. Textfield(a) → InputBox1_2
  57. Textfield(b) → InputBox2_2
  58. Textfield(c) → InputBox3_2
  59. Textfield(d) → InputBox4_2
  60. SurdText(λ1_1) → text14_1
  61. Determinant(T_2) → Determinant_2
  62. Textfield(a) → InputBox1_3
  63. Textfield(b) → InputBox2_3
  64. Textfield(c) → InputBox3_3
  65. Textfield(d) → InputBox4_3
  66. LaTeX(CEigenvalues) → text16