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3. Nonlinear Systems of Two Ordinary Differential Equations

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

  1. _x_′ = x n F(x, y), _y_′ = g(y) F(x, y).
  2. _x_′ = e_λ_x F(x, y), _y_′ = g(y) F(x, y).
  3. _x_′ = F(x, y), _y_′ = G(x, y).Autonomous system of general form.
  4. _x_′ = _f_1(x) _g_1(y) Φ(x, y, t), _y_′ = _f_2(x) _g_2(y) Φ(x, y, t).
  5. x = _tx_′ + F(_x_′, _y_′), y = _ty_′ + G(_x_′, _y_′).Clairaut system.

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

  1. _x_″ = xf(ax − by) + g(ax − by), _y_″ = yf(ax − by) + h(ax − by).
  2. _x_″ = xf(y/x), _y_″ = yg(y/x).
  3. _x_″ = _kxr_−3, _y_″ = _kyr_−3; r = (_x_2 + _y_2)1/2.Equations of motion of a point mass in gravitational field.
  4. _x_″ = xf(r), _y_″ = yf(r); r = (_x_2 + _y_2)1/2.Equations of motion of a point mass in central force field.
  5. _x_″ = xf(_x_2 + _y_2, y/x) − yg(y/x), _y_″ = yf(_x_2 + _y_2, y/x) + xg(y/x).
  6. x_″ = −_f(y)g(v)_x_′, y_″ = −_f(y)g(v)_y_′ − a; v = [(_x_′)2 + (_y_′)2]1/2.Equations of motion of a projectile.
  7. _x_″ + a(t)x = x_−3_f(y/x), _y_″ + a(t)y = y_−3_g(y/x).Generalized Ermakov (Yermakov) system.
  8. _x_″ = x_−3_F(x/φ(t), y/φ(t)), _y_″ = y_−3_G(x/φ(t), y/φ(t)); φ(t) = (_at_2 + bt + c)1/2.
  9. _x_″ = f(_y_′/_x_′), _y_″ = g(_y_′/_x_′).
  10. _x_″ = _x_Φ(x, y, t, _x_′, _y_′), _y_″ = _y_Φ(x, y, t, _x_′, _y_′).
  11. _x_″ + x_−3_f(y/x) =_x_Φ(x, y, t, _x_′, _y_′), _y_″ + y_−3_g(y/x) =_y_Φ(x, y, t, _x_′, _y_′).
  12. _x_″ = F(t, _tx_′ − x, _ty_′ − y), _y_″ = G(t, _tx_′ − x, _ty_′ − y).
  13. _x_″ = _x_′Φ(x, y, t, _x_′, _y_′) + f(y), _y_″ = −_y_′Φ(x, y, t, _x_′, _y_′) + g(x).
  14. _x_″ = _ay_′Φ(x, y, t, _x_′, _y_′) + f(x), _y_″ = _bx_′Φ(x, y, t, _x_′, _y_′) + g(y).
  15. _x_″ = f(_y_′)Φ(x, y, t, _x_′, _y_′), _y_″ = g(_x_′)Φ(x, y, t, _x_′, _y_′).

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