Solve by finding first integrals the symmetric system…
Solve by finding first integrals the symmetric system: $$ \frac{dx}{-x^2} = \frac{dy}{xy-2z^2} = \frac{dz}{xz} $$
Explanation of Method
The first and third ratios combine into the exact differential $d(xz)=0$, giving an immediate first integral; substituting $z=C_1/x$ into the second ratio produces a linear ODE for $y(x)$, solved with an integrating factor.
- Equate the first and third ratios to find an exact differential.
- Integrate to get the first integral $xz=C_1$.
- Substitute into the second ratio to obtain a linear ODE for $y(x)$.
- Solve with an integrating factor to find the second integral.
Step 1 · First integral
Equating the first and third fractions, $xz\,dx=-x^2\,dz$, i.e. $z\,dx+x\,dz=0$. This is $d(xz)=0$, so
$$ xz=C_1. $$
Step 2 · Linear ODE for $y(x)$
Equating the first and second fractions and substituting $z=C_1/x$ gives
$$ \frac{dy}{dx}+\frac{y}{x}=\frac{2C_1^2}{x^4}. $$
Step 3 · Solve with an integrating factor
With $\mu=x$, $\dfrac{d}{dx}(xy)=\dfrac{2C_1^2}{x^3}$, so $xy=-\dfrac{C_1^2}{x^2}+C_2$. Substituting $C_1^2/x^2=z^2$,
$$ xy+z^2=C_2. $$
Step 4 · Verification
Differentiating $xz$ and $xy+z^2$ along the flow $\dot x=-x^2,\ \dot y=xy-2z^2,\ \dot z=xz$ confirms both are conserved. ✓
Summary Table
| Quantity | Value |
|---|---|
| First integral | $xz=C_1$ |
| Reduced linear ODE | $dy/dx+y/x=2C_1^2/x^4$ |
| Second integral | $xy+z^2=C_2$ |
Additional Notes
- The first and third fractions combine into a perfect differential $d(xz)$ without any further manipulation — the fastest possible route to a first integral.
- The same combination $xy+z^2$ also appeared as a second integral in Problem 21, though the two systems are different; both stem from the same recurring "quadratic form is conserved" pattern.
- Verified by direct differentiation of both conserved quantities along the system.
Result
The first integrals of the system are: