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Euler Solution of ODE

Euler Solution of ODE
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2014-11-07 03:52
FunctionCalculusDerivative
This example demonstrates the Euler method (forward Euler) for solving a simple ordinary differential equation (ODE). Students can learn that there are two variables influencing the solution: τ. By adjusting the input boxes, one can observe the changes in the solution. The core mathematical concepts involved include ordinary differential equations, difference and slope. The GeoGebra commands used include: Textfield, Line, Segment; objects: τ, E, ode, inputBox1, init, inputBox2, Init, a, u_0, u_1, h, f, inputBox3, step, b, A, d, c, B, e, g, C, i, j, D, k, l, F, m, n, G, p, q, H, r, s, I, t, a_1, J. Suitable for high school mathematics education.

GeoGebra Commands & Parameters

  1. Textfield(E) → inputBox1
  2. Textfield(init) → inputBox2
  3. Line((0, E), xAxis) → a
  4. Textfield(h) → inputBox3
  5. Line(Init, (x(Init) + h, f(y(Init)))) → b
  6. Segment(Init, A) → d
  7. Line(A, (x(A) + h, f(y(A)))) → c
  8. Segment(A, B) → e
  9. Line(B, (x(B) + h, f(y(B)))) → g
  10. Segment(B, C) → i
  11. Line(C, (x(C) + h, f(y(C)))) → j
  12. Segment(C, D) → k
  13. Line(D, (x(D) + h, f(y(D)))) → l
  14. Segment(D, F) → m
  15. Line(F, (x(F) + h, f(y(F)))) → n
  16. Segment(F, G) → p
  17. Line(G, (x(G) + h, f(y(G)))) → q
  18. Segment(G, H) → r
  19. Line(H, (x(H) + h, f(y(H)))) → s
  20. Segment(H, I) → t
  21. Line(I, (x(I) + h, f(y(I)))) → a_1
  22. Segment(I, J) → b_1
  23. Line(J, (x(J) + h, f(y(J)))) → c_1
  24. Segment(J, K) → d_1
  25. Line(K, (x(K) + h, f(y(K)))) → e_1
  26. Segment(K, L) → f_1
  27. Line(L, (x(L) + h, f(y(L)))) → g_1
  28. Segment(L, M) → h_1
  29. Line(M, (x(M) + h, f(y(M)))) → i_1
  30. Segment(M, N) → j_1
  31. Line(N, (x(N) + h, f(y(N)))) → k_1
  32. Segment(N, O) → l_1
  33. Line(O, (x(O) + h, f(y(O)))) → m_1
  34. Segment(O, P) → n_1
  35. Line(P, (x(P) + h, f(y(P)))) → p_1
  36. Segment(P, Q) → q_1
  37. Line(Q, (x(Q) + h, f(y(Q)))) → r_1
  38. Segment(Q, R) → s_1
  39. Line(R, (x(R) + h, f(y(R)))) → t_1
  40. Segment(R, S) → a_2
  41. Line(S, (x(S) + h, f(y(S)))) → b_2
  42. Segment(S, T) → c_2
  43. Line(T, (x(T) + h, f(y(T)))) → d_2
  44. Segment(T, U) → e_2
  45. Line(U, (x(U) + h, f(y(U)))) → f_2
  46. Segment(U, V) → g_2
  47. Line(V, (x(V) + h, f(y(V)))) → h_2
  48. Segment(V, W) → i_2
  49. Line(W, (x(W) + h, f(y(W)))) → j_2
  50. Segment(W, Z) → k_2
  51. Line(Z, (x(Z) + h, f(y(Z)))) → l_2
  52. Segment(Z, A_1) → m_2
  53. Line(A_1, (x(A_1) + h, f(y(A_1)))) → n_2
  54. Segment(A_1, B_1) → p_2
  55. Line(B_1, (x(B_1) + h, f(y(B_1)))) → q_2
  56. Segment(B_1, C_1) → r_2
  57. Line(C_1, (x(C_1) + h, f(y(C_1)))) → s_2
  58. Segment(C_1, D_1) → t_2
  59. Line(D_1, (x(D_1) + h, f(y(D_1)))) → a_3
  60. Segment(D_1, E_1) → b_3
  61. Line(E_1, (x(E_1) + h, f(y(E_1)))) → c_3
  62. Segment(E_1, F_1) → d_3
  63. Line(F_1, (x(F_1) + h, f(y(F_1)))) → e_3
  64. Segment(F_1, G_1) → f_3
  65. Line(G_1, (x(G_1) + h, f(y(G_1)))) → g_3
  66. Segment(G_1, H_1) → h_3
  67. Line(H_1, (x(H_1) + h, f(y(H_1)))) → i_3
  68. Segment(H_1, I_1) → j_3
  69. Line(I_1, (x(I_1) + h, f(y(I_1)))) → k_3
  70. Segment(I_1, J_1) → l_3
  71. Line(J_1, (x(J_1) + h, f(y(J_1)))) → m_3
  72. Segment(J_1, K_1) → n_3
  73. Line(K_1, (x(K_1) + h, f(y(K_1)))) → p_3
  74. Segment(K_1, L_1) → q_3
  75. Line(L_1, (x(L_1) + h, f(y(L_1)))) → r_3
  76. Segment(L_1, M_1) → s_3
  77. Line(M_1, (x(M_1) + h, f(y(M_1)))) → t_3
  78. Segment(M_1, N_1) → a_4
  79. Line(N_1, (x(N_1) + h, f(y(N_1)))) → b_4
  80. Segment(N_1, O_1) → c_4
  81. Line(O_1, (x(O_1) + h, f(y(O_1)))) → d_4
  82. Segment(O_1, P_1) → e_4
  83. Line(P_1, (x(P_1) + h, f(y(P_1)))) → f_4
  84. Segment(P_1, Q_1) → g_4
  85. Line(Q_1, (x(Q_1) + h, f(y(Q_1)))) → h_4
  86. Segment(Q_1, R_1) → i_4
  87. Line(R_1, (x(R_1) + h, f(y(R_1)))) → j_4
  88. Segment(R_1, S_1) → k_4
  89. Line(S_1, (x(S_1) + h, f(y(S_1)))) → l_4
  90. Segment(S_1, T_1) → m_4
  91. Line(T_1, (x(T_1) + h, f(y(T_1)))) → n_4
  92. Segment(T_1, U_1) → p_4
  93. Line(U_1, (x(U_1) + h, f(y(U_1)))) → q_4
  94. Segment(U_1, V_1) → r_4
  95. Line(V_1, (x(V_1) + h, f(y(V_1)))) → s_4
  96. Segment(V_1, W_1) → t_4
  97. Line(W_1, (x(W_1) + h, f(y(W_1)))) → a_5
  98. Segment(W_1, Z_1) → b_5
  99. Line(Z_1, (x(Z_1) + h, f(y(Z_1)))) → c_5
  100. Segment(Z_1, A_2) → d_5
  101. Line(A_2, (x(A_2) + h, f(y(A_2)))) → e_5
  102. Segment(A_2, B_2) → f_5
  103. Line(B_2, (x(B_2) + h, f(y(B_2)))) → g_5
  104. Segment(B_2, C_2) → h_5