Solve by finding first integrals the symmetric system…
Solve by finding first integrals the symmetric system: $$ \frac{dx}{x(y^2-z^2)} = \frac{dy}{-y(z^2+x^2)} = \frac{dz}{z(x^2+y^2)} $$
Explanation of Method
Adding the logarithmic rates $\dot y/y+\dot z/z$ reproduces exactly $\dot x/x$, giving a multiplicative first integral; multiplying each equation by its own variable and adding shows the nonlinear terms cancel completely, giving a conserved Euclidean norm.
- Add the logarithmic rates of $y$ and $z$ and compare with that of $x$.
- Integrate the resulting relation to get the first integral.
- Multiply each equation by its variable and add to eliminate the cubic terms.
- Integrate to get the spherical second integral.
Step 1 · First integral
With common ratio $d\tau$, $\dfrac{dy}{y}+\dfrac{dz}{z}=\bigl[-(z^2+x^2)+(x^2+y^2)\bigr]d\tau=(y^2-z^2)\,d\tau=\dfrac{dx}{x}$, so
$$ \frac{dx}{x}-\frac{dy}{y}-\frac{dz}{z}=0 \implies \frac{x}{yz}=C_1. $$
Step 2 · Spherical integral
Multiplying each equation by its own variable and adding,
$$ x\dot x+y\dot y+z\dot z=x^2(y^2-z^2)-y^2(z^2+x^2)+z^2(x^2+y^2)=0, $$
so
$$ x^2+y^2+z^2=C_2. $$
Step 3 · Verification
Differentiating $x/(yz)$ and $x^2+y^2+z^2$ along the flow shows both quantities are exactly conserved, with all cubic cross-terms cancelling. ✓
Summary Table
| Quantity | Value |
|---|---|
| First integral | $x/(yz)=C_1$ |
| Second integral | $x^2+y^2+z^2=C_2$ |
| Geometric picture | trajectory = sphere ∩ cubic surface |
Additional Notes
- This system has the same structural fingerprint as Problem 18 — both conserve a quadratic norm and a ratio — a signature of rotation-like nonlinear couplings.
- Neither integral requires solving for $x(\tau), y(\tau), z(\tau)$ explicitly; the nonlinear cubic terms cancel algebraically in both combinations.
- Verified by direct differentiation of each conserved quantity along the flow.
Result
The first integrals of the system are: