Solve the system of differential equations: dot x=x+y+z, dot…
Solve the system of differential equations: $$ \dot x=x+y+z,\qquad \dot y=y+z,\qquad \dot z=x $$
Explanation of Method
The eigenvalue $\lambda=0$ has algebraic multiplicity 2 but geometric multiplicity only 1, so a genuine Jordan block appears; a generalized eigenvector produces a linear-in-$t$ secular term alongside the growing $e^{2t}$ mode from the simple eigenvalue.
- Write the system in matrix form and compute the characteristic polynomial.
- Identify the repeated eigenvalue $\lambda=0$ and the simple eigenvalue $\lambda=2$.
- Find the eigenvector at $\lambda=0$ and confirm its geometric multiplicity is only 1.
- Solve for a generalized eigenvector and build the Jordan-chain solution.
- Assemble and verify the general solution.
Step 1 · Matrix form and characteristic equation
With $A=\begin{pmatrix}1&1&1\\0&1&1\\1&0&0\end{pmatrix}$, expanding $\det(A-\lambda I)$ gives
$$ \lambda^2(2-\lambda)=-\lambda^2(\lambda-2)=0. $$
Eigenvalues: $\lambda_1=0$ (algebraic multiplicity 2) and $\lambda_2=2$ (simple).
Step 2 · Eigenvector for $\lambda_2=2$
$$ \mathbf v_3=\begin{pmatrix}2\\1\\1\end{pmatrix},\qquad A\mathbf v_3=2\mathbf v_3.\ \checkmark $$
Step 3 · Eigenvector for $\lambda_1=0$ and its rank deficiency
$A$ has rank 2 (rows $(1,1,1)$ and $(0,1,1)$ are independent), so the null space is one-dimensional:
$$ \mathbf p=\begin{pmatrix}0\\-1\\1\end{pmatrix},\qquad A\mathbf p=\mathbf 0.\ \checkmark $$
Geometric multiplicity $1<2$: a generalized eigenvector is required.
Step 4 · Generalized eigenvector and Jordan-chain solutions
Solving $A\mathbf w=\mathbf p$ gives $\mathbf w=(1,-1,0)$, verified directly. The three independent solutions are
$$ \mathbf u^{(1)}=\mathbf p,\qquad \mathbf u^{(2)}=t\,\mathbf p+\mathbf w,\qquad \mathbf u^{(3)}=e^{2t}\mathbf v_3. $$
Step 5 · General solution and verification
The three solutions are independent at $t=0$ ($\det=4\neq0$); direct substitution confirms the general solution below satisfies all three original equations. ✓
Summary Table
| Quantity | Value |
|---|---|
| Eigenvalues | $\lambda=0$ (mult.\ 2, geometric mult.\ 1), $\lambda=2$ (simple) |
| Generalized eigenvector | $\mathbf w=(1,-1,0)$ |
| Structure | one $2\times2$ Jordan block at $\lambda=0$ |
Additional Notes
- The relation $\dot z=x$ reflects the nilpotent Jordan structure at $\lambda=0$: the secular term $C_2(1,-(t+1),t)^\top$ grows only linearly, not exponentially.
- As $t\to\infty$ the solution is dominated by $e^{2t}$ unless $C_3=0$.
- Verified by direct substitution into all three original equations.
Result
The general solution of the system is: