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

PDF version of this page

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′ = (b − c)yz,   by′ = (c − a)zx,   cz′ = (a − b)xy.
2. ax′ = (b − c)yzf(x, y, z, t),   by′ = (c − a)zxf(x, y, z, t),   cz′ = (a − b)xyf(x, y, z, t).
3. x′ = a(y − x),   y′ = bx − y − xz,   z′ = −cz + xy.    Lorenz equations.
4. x′ = cF₂ − bF₃,   y′ = aF₃ − cF₁,   z′ = bF₁ − aF₂,   where   F_n = F_n(x, y, z, t).
5. x′ = czF₂ − byF₃,   y′ = axF₃ − czF₁,   z′ = byF₁ − axF₂,   where   F_n = F_n(x, y, z, t).
6. x′ = x(cF₂ − bF₃),   y′ = y(aF₃ − cF₁),   z′ = z(bF₁ − aF₂),   where   F_n = F_n(x, y, z, t).
7. x′ = h(z)F₂ − g(y)F₃,   y′ = f(x)F₃ − h(z)F₁,   z′ = g(y)F₁ − f(x)F₂   where   F_n = F_n(x, y, z, t).

7a. Systems of nonlinear ODEs with homogeneous right-hand sides.

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

8. x″ = F_x,   y″ = F_y,   z″ = F_z,   where   F = F(r),   r = (x² + y² + z²)^1/2.
9. x″ = xF,   y″ = yF,   z″ = zF,   where   F = F(x, y, z, t, x′, y′, z′).
10. x″ = F₁,   y″ = F₂,   z″ = F₃,   where   F_n = F_n(t, tx′ − x, ty′ − y, tz′ − z).
11. x″ = cF₂ − bF₃,   y″ = aF₃ − cF₁,   z″ = bF₁ − aF₂,   where   F = F(x, y, z, t, x′, y′, z′).