Solve by finding first integrals the symmetric system…
Solve by finding first integrals the symmetric system: $$ \frac{dx}{xz} = \frac{dy}{yz} = \frac{dz}{-xy} $$
Explanation of Method
Equating the first two ratios cancels the common factor $z$ and integrates directly to a constant ratio $x/y$; substituting that ratio into the remaining combination produces an exact differential for the second integral.
- Equate the first two ratios and cancel $z$ to get the first integral.
- Equate the first (or second) ratio with the third to get $x\,dy+z\,dz=0$.
- Substitute $x=C_1y$ to integrate this exact combination.
- Verify both integrals are conserved along the flow.
Step 1 · First integral
Equating the first two fractions, $\dfrac{dx}{xz}=\dfrac{dy}{yz}$ gives $\dfrac{dx}{x}=\dfrac{dy}{y}$. Integrating,
$$ \frac{x}{y}=C_1. $$
Step 2 · An exact combination
Equating either the first or second fraction with the third gives, after simplification, the same relation
$$ x\,dy+z\,dz=0. $$
Step 3 · Second integral
Using $x=C_1y$, this becomes $C_1y\,dy+z\,dz=0$, i.e. $d\!\left(\tfrac{C_1y^2}{2}+\tfrac{z^2}{2}\right)=0$. Substituting back $C_1=x/y$,
$$ xy+z^2=C_2. $$
Step 4 · Verification
Differentiating $x/y$ and $xy+z^2$ along the flow $\dot x=xz,\ \dot y=yz,\ \dot z=-xy$ shows both total derivatives vanish identically. ✓
Summary Table
| Quantity | Value |
|---|---|
| First integral | $x/y=C_1$ |
| Key exact combination | $x\,dy+z\,dz=0$ |
| Second integral | $xy+z^2=C_2$ |
Additional Notes
- The first ratio $x/y$ is conserved because $x$ and $y$ pick up the same multiplicative rate $z$ along the flow.
- Substituting the first integral into the leftover exact differential is the standard trick for finding a second integral in symmetric systems.
- Both integrals were verified directly by differentiating along the system.
Result
The first integrals of the system are: