Solve the system of differential equations: (dx)/(y-z) =…
Solve the system of differential equations: $$ \frac{dx}{y-z} = \frac{dy}{z-x} = \frac{dz}{y-x} $$
Explanation of Method
Choosing constant multipliers $(1,1,-1)$ makes the combined denominator vanish identically, giving an immediate first integral $x+y-z=C_1$; substituting that relation into the remaining ratio produces an exact differential whose potential function is the second first integral.
- Search for constant multipliers $\alpha,\beta,\gamma$ making $\alpha P+\beta Q+\gamma R=0$.
- Integrate the resulting exact combination to get the first first integral.
- Use it to eliminate $z$ from the remaining ratio and check exactness.
- Integrate the exact differential to get the second first integral.
Step 1 · First integral
With $P=y-z$, $Q=z-x$, $R=y-x$, the combination $P+Q-R=0$ identically, so $dx+dy-dz=0$:
$$ x+y-z=C_1. $$
Step 2 · Reduce the remaining ratio
Substituting $z=x+y-C_1$ into $\dfrac{dx}{y-z}=\dfrac{dy}{z-x}$ gives
$$ \frac{dx}{C_1-x}=\frac{dy}{y-C_1}\ \Longrightarrow\ (y-C_1)\,dx+(x-C_1)\,dy=0, $$
which is exact since $\partial_y(y-C_1)=\partial_x(x-C_1)=1$.
Step 3 · Second integral
The potential function is $F=xy-C_1(x+y)$; substituting $C_1=x+y-z$ and simplifying gives
$$ x^2+xy+y^2-z(x+y)=C_2. $$
Step 4 · Verification
Differentiating both $\Phi_1=x+y-z$ and $\Phi_2=x^2+xy+y^2-z(x+y)$ along the system and expanding shows each total derivative vanishes identically. ✓
Summary Table
| Quantity | Value |
|---|---|
| Multiplier combination | $(\alpha,\beta,\gamma)=(1,1,-1)$ |
| First integral | $x+y-z=C_1$ |
| Second integral | $x^2+xy+y^2-z(x+y)=C_2$ |
Additional Notes
- Finding the right constant multipliers is largely trial and error, but the pattern $\alpha P+\beta Q+\gamma R\equiv0$ (independent of $x,y,z$) is exactly what to look for in symmetric systems with linear denominators.
- Once one first integral eliminates a variable, the remaining two-variable ratio often becomes an exact differential equation, as it does here.
- Both integrals were verified by expanding their total derivatives along the system and confirming exact cancellation.
Result
The general solution of the system is: