Solve by finding first integrals the symmetric system…
Solve by finding first integrals the symmetric system: $$ \frac{dx}{y+z} = \frac{dy}{x+z} = \frac{dz}{x+y} $$
Explanation of Method
Multiplier combinations $(1,-1,0)$ and $(0,1,-1)$ show that $x-y$ and $y-z$ decay at the same relative rate, giving a clean ratio integral; comparing the decay rate of $x-y$ to the growth rate of the sum $x+y+z$ (which is exactly twice as fast) produces a second integral with a squared factor.
- Use multipliers $(1,-1,0)$ and $(0,1,-1)$ to show $x-y$ and $y-z$ satisfy the same relative rate equation.
- Integrate to get the ratio first integral.
- Compare the rate of $x-y$ with the rate of the sum $x+y+z$.
- Integrate the resulting relation, noting the factor-of-two mismatch requires a squared term.
Step 1 · First integral
Since $(y+z)-(x+z)=y-x$ and $(x+z)-(x+y)=z-y$, the combinations give $\dfrac{d(x-y)}{x-y}=\dfrac{d(y-z)}{y-z}$. Integrating,
$$ \frac{x-y}{y-z}=C_1. $$
Step 2 · Rate of the sum
Adding all three equations, $\dfrac{d(x+y+z)}{x+y+z}=2\,d\tau$, while $\dfrac{d(x-y)}{x-y}=-d\tau$ — the sum grows exactly twice as fast (in log-rate) as $x-y$ decays.
Step 3 · Second integral
Combining $2\ln|x-y|+\ln|x+y+z|=\text{const}$ gives
$$ (x-y)^2(x+y+z)=C_2. $$
Step 4 · Verification
Differentiating $\Phi_2=(x-y)^2(x+y+z)$ using $d(x-y)/d\tau=-(x-y)$ and $d(x+y+z)/d\tau=2(x+y+z)$ gives $-2(x-y)^2(x+y+z)+2(x-y)^2(x+y+z)=0$. ✓
Summary Table
| Quantity | Value |
|---|---|
| First integral | $(x-y)/(y-z)=C_1$ |
| Rate of $x-y$ | $-1$ (relative) |
| Rate of $x+y+z$ | $+2$ (relative) |
| Second integral | $(x-y)^2(x+y+z)=C_2$ |
Additional Notes
- The naive guess $(x-y)(x+y+z)=\text{const}$ fails because the two quantities decay/grow at different relative rates (ratio $1:2$); matching the rates requires squaring $(x-y)$.
- By symmetry, $(y-z)^2(x+y+z)$ is an equally valid second integral, related to the one above by $C_1^2$.
- Both integrals were verified by differentiating along the flow and confirming exact cancellation.
Result
The first integrals of the system are: