Solve by finding first integrals the system: (dx)/(dt) =…
Solve by finding first integrals the system: $$ \frac{dx}{dt} = \frac{x-y}{z-t},\qquad \frac{dy}{dt} = \frac{x-y}{z-t},\qquad \frac{dz}{dt} = x-y+1 $$
Explanation of Method
The right-hand sides of the $x$- and $y$-equations are identical, so their difference is immediately conserved; substituting that constant simplifies the $z$-equation to a linear ODE in $t$, and one further integration recovers $x(t)$ explicitly via a logarithm.
- Subtract the $y$-equation from the $x$-equation to find the first integral.
- Substitute the constant $x-y$ into the $z$-equation and integrate.
- Use the resulting relation to reduce the $x$-equation to a separable ODE.
- Integrate once more to obtain $x(t)$, then recover $y(t)$ and $z(t)$.
Step 1 · First integral
Since $\dot x$ and $\dot y$ have the same right-hand side, $\dfrac{d(x-y)}{dt}=0$, so
$$ x-y=C_1. $$
Step 2 · Second integral
With $x-y=C_1$, the third equation becomes $\dot z=C_1+1$, so $z=(C_1+1)t+\text{const}$, giving
$$ z-(x-y+1)\,t=C_2. $$
Step 3 · Third integral
Since $z-t=C_1t+C_2$, the first equation becomes $\dfrac{dx}{dt}=\dfrac{C_1}{C_1t+C_2}$. Integrating,
$$ x-\ln|z-t|=C_3. $$
Step 4 · Explicit solution
$$ x(t)=C_3+\ln|C_1t+C_2|,\qquad y(t)=x(t)-C_1,\qquad z(t)=(C_1+1)t+C_2. $$
Step 5 · Verification
Substituting back confirms all three original equations, as well as $x-y=C_1$, $z-(x-y+1)t=C_2$, and $x-\ln|z-t|=C_3$, hold identically. ✓
Summary Table
| Quantity | Value |
|---|---|
| First integral | $x-y=C_1$ |
| Second integral | $z-(x-y+1)t=C_2$ |
| Third integral | $x-\ln|z-t|=C_3$ |
Additional Notes
- Because $\dot x\equiv\dot y$, the difference $x-y$ is conserved with no integration required — the simplest possible first integral.
- The special case $C_1=0$ (i.e. $x=y$) makes $\dot x=0$ as well, so $x$ is constant and the logarithmic term in $\Phi_3$ still reduces consistently.
- Verified by direct substitution of the explicit closed forms into all three original equations.
Result
The general solution of the system is: