Solve the system of differential equations: dot x =…
Solve the system of differential equations: $$ \dot x = x(y^2-z^2),\qquad \dot y = -y(x^2+z^2),\qquad \dot z = z(x^2+y^2) $$
Explanation of Method
Dividing each equation by its variable and combining logarithmic differentials cancels the nonlinear terms exactly, yielding a first integral $x/(yz)=C_1$; multiplying instead by $x,y,z$ and adding shows the sum $x\,dx+y\,dy+z\,dz$ vanishes identically, giving a conserved sphere radius.
- Divide each equation by its variable to get logarithmic differentials.
- Add $dy/y+dz/z$ and compare with $dx/x$ to find the first (multiplicative) integral.
- Form $x\,dx+y\,dy+z\,dz$ and show it vanishes identically to get the second (spherical) integral.
- Verify both are conserved along the flow.
Step 1 · Logarithmic combination
Since $\dfrac{dy}{y}+\dfrac{dz}{z}=\bigl[-(x^2+z^2)+(x^2+y^2)\bigr]dt=(y^2-z^2)\,dt=\dfrac{dx}{x}$,
$$ \frac{dx}{x}-\frac{dy}{y}-\frac{dz}{z}=0\ \Longrightarrow\ \frac{x}{yz}=C_1. $$
Step 2 · Spherical integral
Multiplying each equation by its own variable and adding,
$$ x\dot x+y\dot y+z\dot z=x^2(y^2-z^2)-y^2(x^2+z^2)+z^2(x^2+y^2)=0, $$
so
$$ x^2+y^2+z^2=C_2. $$
Step 3 · General solution
Every trajectory lies on the intersection of the sphere $x^2+y^2+z^2=C_2$ and the surface $x=C_1yz$.
Step 4 · Verification
Direct differentiation confirms $\dfrac{d}{dt}\!\left(\dfrac{x}{yz}\right)=0$ and $\dfrac{d}{dt}(x^2+y^2+z^2)=0$ along the flow. ✓
Summary Table
| Quantity | Value |
|---|---|
| First integral | $x/(yz)=C_1$ |
| Second integral | $x^2+y^2+z^2=C_2$ |
| Geometric picture | trajectory = sphere ∩ cubic surface |
Additional Notes
- The identity $x\,dx+y\,dy+z\,dz\equiv0$ along the flow is a strong hint that this system conserves the Euclidean norm — a common structure in rotation-like nonlinear systems.
- Both integrals were found without ever solving for $x(t),y(t),z(t)$ explicitly, since the nonlinear terms cancel algebraically.
- Verified by direct differentiation of each conserved quantity along the system.
Result
The general solution of the system is: