Systems of Ordinary Differential Equations (original) (raw)

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1.1. Systems of First-Order Ordinary Differential Equations;x = x(t), y = y(t)

  1. _x_′ = ax + by, _y_′ = cx + dy.
  2. _x_′ = a_1_x + b_1_y + _c_1, _y_′ = a_2_x + b_2_y + _c_2.
  3. _x_′ = f(t)x + g(t)y, _y_′ = g(t)x + f(t)y.
  4. _x_′ = f(t)x + g(t)y, y_′ = −_g(t)x + f(t)y.
  5. _x_′ = f(t)x + g(t)y, _y_′ = ag(t)x + [f(t) + bg(t)]y.
  6. _x_′ = f(t)x + g(t)y, _y_′ = a[f(t) + ah(t)]x + a[g(t) − h(t)]y.
  7. _x_′ = f(t)x + g(t)y, _y_′ = h(t)x + p(t)y.

1.2. Systems of Second-Order Ordinary Differential Equations;x = x(t), y = y(t)

  1. _x_″ = ax + by, _y_″ = cx + dy.
  2. _x_″ = a_1_x + b_1_y + _c_1, _y_″ = a_2_x + b_2_y + _c_2.
  3. _x_″ − _ay_′ + bx = 0, _y_″ + _ax_′ + by = 0.
  4. _x_″ + _a_1_x_′ + _b_1_y_′ +c_1_x + d_1_y =k_1_e i_ω_t, _y_″ + _a_2_x_′ + _b_2_y_′ +c_2_x + d_2_y =k_2_e i_ω_t.
  5. _x_″ = a(_ty_′ − y), _y_″ = b(_tx_′ − x).
  6. _x_″ = f(t)(a_1_x + b_1_y), _y_″ = f(t)(a_2_x + b_2_y).
  7. _x_″ = f(t)(_a_1_x_′ + b_1_y′), _y_″ = f(t)(_a_2_x_′ + b_2_y′).
  8. _x_″ = af(t)(_ty_′ − y), _y_″ = bf(t)(_tx_′ − x).
  9. _t_2_x_″ + _a_1_tx_′ + _b_1_ty_′ +c_1_x + d_1_y = 0, _t_2_y_″ + _a_2_tx_′ + _b_2_ty_′ +c_2_x + d_2_y = 0.
  10. (α_t_2 + β_t_ + σ)2_x_″ = ax + by, (α_t_2 + β_t_ + σ)2_y_″ = cx + dy.
  11. _x_″ = f(t)(_tx_′ − x) +g(t)(_ty_′ − y), _y_″ = h(t)(_tx_′ − x) +p(t)(_ty_′ − y).

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