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Exact Solutions > Systems of Ordinary Differential Equations > Nonlinear Systems of Three or More Ordinary Differential Equations

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4. Nonlinear Systems of Three or More Ordinary Differential Equations

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

  1. ax′ = (bc)yz,   by′ = (ca)zx,   cz′ = (ab)xy.
  2. ax′ = (bc)yzf(x, y, z, t),   by′ = (ca)zxf(x, y, z, t),   cz′ = (ab)xyf(x, y, z, t).
  3. x′ = a(yx),   y′ = bxyxz,   z′ = −cz + xy.    Lorenz equations.
  4. x′ = cF2bF3,   y′ = aF3cF1,   z′ = bF1aF2,   where   Fn = Fn(x, y, z, t).
  5. x′ = czF2byF3,   y′ = axF3czF1,   z′ = byF1axF2,   where   Fn = Fn(x, y, z, t).
  6. x′ = x(cF2bF3),   y′ = y(aF3cF1),   z′ = z(bF1aF2),   where   Fn = Fn(x, y, z, t).
  7. x′ = h(z)F2g(y)F3,   y′ = f(x)F3h(z)F1,   z′ = g(y)F1f(x)F2   where   Fn = Fn(x, y, z, t).

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

  1. x″ = Fx,   y″ = Fy,   z″ = Fz,   where   F = F(r),   r = (x2 + y2 + z2)1/2.
  2. x″ = xF,   y″ = yF,   z″ = zF,   where   F = F(x, y, z, t, x′, y′, z′).
  3. x″ = F1,   y″ = F2,   z″ = F3,   where   Fn = Fn(t, tx′ − x, ty′ − y, tz′ − z).
  4. x″ = cF2bF3,   y″ = aF3cF1,   z″ = bF1aF2,   where   F = F(x, y, z, t, x′, y′, z′).

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