Solve by finding first integrals the symmetric system…
Solve by finding first integrals the symmetric system: $$ \frac{dx}{x(y+z)} = \frac{dy}{z(z-y)} = \frac{dz}{y(y-z)} $$
Explanation of Method
The last two ratios combine into the exact relation $y\,dy+z\,dz=0$, giving a rotation-type invariant $y^2+z^2$; a second, independent invariant comes from combining $x$ with $y-z$ directly, since their relative growth rates exactly cancel.
- Equate the last two ratios and factor out $(y-z)$ to get $y\,dy+z\,dz=0$.
- Integrate to find the first (rotational) integral.
- Compute the relative rates of $x$ and $y-z$ along the flow.
- Combine them into a second, independent product integral.
Step 1 · First integral
Equating the second and third fractions, $y(y-z)\,dy=z(z-y)\,dz$; factoring out $(y-z)$ gives $y\,dy+z\,dz=0$. Integrating,
$$ y^2+z^2=C_1. $$
Step 2 · Relative rates of $x$ and $y-z$
With common ratio $d\tau$, $\dot x=x(y+z)$ and $\dot y-\dot z=z(z-y)-y(y-z)=(z-y)(y+z)=-(y-z)(y+z)$, so
$$ \frac{d(x)}{x}=(y+z)\,d\tau,\qquad \frac{d(y-z)}{y-z}=-(y+z)\,d\tau. $$
Step 3 · Second integral
Adding these two logarithmic rates cancels the $(y+z)$ factor exactly:
$$ \frac{d\bigl[x(y-z)\bigr]}{x(y-z)}=\frac{dx}{x}+\frac{d(y-z)}{y-z}=0, $$
so
$$ x(y-z)=C_2. $$
Step 4 · Verification
Differentiating $y^2+z^2$ and $x(y-z)$ directly along the flow shows both total derivatives vanish identically. ✓
Summary Table
| Quantity | Value |
|---|---|
| First integral | $y^2+z^2=C_1$ |
| Relative rate of $x$ | $+(y+z)$ |
| Relative rate of $y-z$ | $-(y+z)$ |
| Second integral | $x(y-z)=C_2$ |
Additional Notes
- The key simplification is noticing that $x$ and $y-z$ have exactly opposite relative growth rates along the flow, so their product is conserved — no exponential or logarithmic terms are needed.
- The first integral $y^2+z^2=C_1$ reflects a rotational symmetry between $y$ and $z$ in the last two equations.
- Both integrals were verified by direct differentiation along the system, giving exact cancellation in each case.
Result
The first integrals of the system are: