Solve the system of differential equations: dot x = y - z…
Solve the system of differential equations: $$ \dot x = y - z,\qquad \dot y = x^2 + y,\qquad \dot z = x^2 + z $$
Explanation of Method
Subtracting the last two equations shows $y-z$ satisfies a simple linear relation with $x$, giving a clean first integral $y-z-x=C_1$; substituting this into a combination of the equations reduces the remaining freedom to a first-order linear ODE, solved with an integrating factor.
- Subtract the second and third equations to relate $y-z$ to $x$.
- Combine with the first equation to get the first integral.
- Use it to write a linear ODE for $y+z$ as a function of $x$.
- Solve with an integrating factor to get the second integral.
Step 1 · First integral
Subtracting, $\dot y-\dot z=y-z$; combined with $\dot x=y-z$ this gives $d(y-z)=dx$, so
$$ y-z-x=C_1. $$
Step 2 · Linear ODE for $y+z$
Writing $p=y+z$ and using $y-z=x+C_1$, the ratio $\dfrac{d(y+z)}{2x^2+y+z}=\dfrac{dx}{x+C_1}$ becomes the linear ODE
$$ \frac{dp}{dx}-\frac{p}{x+C_1}=\frac{2x^2}{x+C_1}. $$
Step 3 · Solve with an integrating factor
Using $\mu=1/(x+C_1)$ and integrating, the second first integral is
$$ \frac{y+z}{y-z}-2x+4(y-z-x)\ln|y-z|+\frac{2(y-z-x)^2}{y-z}=C_2. $$
Step 4 · Verification
Differentiating $\Phi_1=y-z-x$ along the system gives $\dot\Phi_1=(y-z)-(y-z)=0$, confirming the first integral. The special case $C_1=0$ (i.e. $y-z=x$) reduces to the explicit solution $y+z=2x^2+C_2x$, verified directly. ✓
Summary Table
| Quantity | Value |
|---|---|
| First integral | $y-z-x=C_1$ |
| Special case $C_1=0$ | $y+z=2x^2+C_2x$ |
| Second integral (general) | involves $\ln|y-z|$ |
Additional Notes
- The first integral $y-z-x=C_1$ is the key simplification — it converts the nonlinear coupling into a single parameter that makes the remaining equation linear in $p=y+z$.
- For the special case $C_1=0$, the second integral collapses to a clean polynomial relation; the general case retains a logarithmic term.
- The first integral was verified directly; the special-case solution was verified by substitution into the original system.
Result
The general solution of the system is: