Solve by finding first integrals the symmetric four-variable…
Solve by finding first integrals the symmetric four-variable system: $$ \frac{dx}{y-u} = \frac{dy}{z-x} = \frac{dz}{u-y} = \frac{du}{x-z} $$
Explanation of Method
Pairing opposite equations shows $x+z$ and $y+u$ are each conserved; the sum of squares is conserved as well (a hidden rotational symmetry), and substituting the two linear integrals collapses the system to a simple harmonic oscillator.
- Add the first and third equations, and the second and fourth, to find two conserved sums.
- Multiply each equation by its own variable and add to find a conserved sum of squares.
- Substitute $z=C_1-x$, $u=C_2-y$ to reduce to a two-variable system.
- Differentiate once more to get a harmonic-oscillator ODE and solve it.
Step 1 · Two linear first integrals
With $\dot x=y-u$, $\dot z=u-y$, adding gives $\dot x+\dot z=0$; similarly $\dot y+\dot u=0$. Hence
$$ x+z=C_1,\qquad y+u=C_2. $$
Step 2 · Conserved sum of squares
Multiplying each equation by its variable and adding, $x\dot x+y\dot y+z\dot z+u\dot u=0$ identically, so
$$ x^2+y^2+z^2+u^2=C_3. $$
Step 3 · Reduce to two variables
Substituting $z=C_1-x$, $u=C_2-y$ gives $\dot x=2y-C_2$, $\dot y=C_1-2x$. Differentiating the first and substituting the second yields
$$ \ddot x+4x=2C_1. $$
Step 4 · Solve the oscillator
This is a harmonic oscillator with constant forcing, giving $x=\tfrac{C_1}{2}+A\cos2\tau+B\sin2\tau$, and correspondingly for $y,z,u$.
Step 5 · Verification
Substituting the explicit forms back into all four original equations, and into $\Phi_3=x^2+y^2+z^2+u^2$, confirms every relation holds identically. ✓
Summary Table
| Quantity | Value |
|---|---|
| Linear integrals | $x+z=C_1,\ y+u=C_2$ |
| Spherical integral | $x^2+y^2+z^2+u^2=C_3$ |
| Reduced ODE | $\ddot x+4x=2C_1$ (harmonic, frequency 2) |
Additional Notes
- Only three of the four naive integrals ($\Phi_1,\Phi_2,\Phi_3$, and the cross-term $xz+yu$) are independent, since $C_1^2+C_2^2=C_3+2(xz+yu)$.
- The reduction to a harmonic oscillator reveals that the trajectory is a circular rotation in the $(x,z)$ and $(y,u)$ planes at the same frequency.
- Verified by direct substitution of the explicit trigonometric solution into all four original equations.
Result
The general solution of the system is: