Solve the system of differential equations: dot x=x-y+z, dot…
Solve the system of differential equations: $$ \dot x=x-y+z,\qquad \dot y=x+y-z,\qquad \dot z=-y+2z $$
Explanation of Method
The eigenvalue $\lambda=1$ has algebraic multiplicity 2 but only a one-dimensional eigenspace, so a genuine Jordan block appears: a generalized eigenvector $\mathbf w$ solving $(A-I)\mathbf w=\mathbf v$ is required to complete a three-dimensional solution basis, producing a $te^t$ secular term.
- Write the system in matrix form and compute the characteristic polynomial.
- Identify the simple eigenvalue $\lambda=2$ and the repeated eigenvalue $\lambda=1$.
- Find the eigenvector at $\lambda=1$ and confirm the 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 $A=\begin{pmatrix}1&-1&1\\1&1&-1\\0&-1&2\end{pmatrix}$, expanding $\det(A-\lambda I)$ gives
$$ -(\lambda-2)(\lambda-1)^2=0. $$
Eigenvalues: $\lambda_1=1$ (algebraic multiplicity 2) and $\lambda_2=2$ (simple).
Step 2 · Eigenvector for $\lambda_2=2$
$$ \mathbf v_1=\begin{pmatrix}1\\0\\1\end{pmatrix},\qquad A\mathbf v_1=2\mathbf v_1.\ \checkmark $$
Step 3 · Eigenvector for $\lambda_1=1$ and its rank deficiency
$A-I=\begin{pmatrix}0&-1&1\\1&0&-1\\0&-1&1\end{pmatrix}$ has rank 2 (rows 1 and 3 are identical), so the null space is only one-dimensional:
$$ \mathbf v_2=\begin{pmatrix}1\\1\\1\end{pmatrix},\qquad A\mathbf v_2=\mathbf v_2.\ \checkmark $$
Geometric multiplicity $1<2$: a generalized eigenvector is required.
Step 4 · Generalized eigenvector and Jordan-chain solutions
Solving $(A-I)\mathbf w=\mathbf v_2$ (consistent since $\mathbf v_2$ lies in the range of $A-I$) gives $\mathbf w=(1,-1,0)$, verified directly. The Jordan chain produces three independent solutions:
$$ \mathbf u^{(1)}=e^{2t}\mathbf v_1,\qquad \mathbf u^{(2)}=e^{t}\mathbf v_2,\qquad \mathbf u^{(3)}=e^{t}(t\,\mathbf v_2+\mathbf w). $$
Step 5 · General solution and verification
The three solutions are linearly independent at $t=0$ ($\det=-1\neq0$); direct substitution confirms the general solution below satisfies all three original equations exactly. ✓
Summary Table
| Quantity | Value |
|---|---|
| Eigenvalues | $\lambda=1$ (mult.\ 2, geometric mult.\ 1), $\lambda=2$ (simple) |
| Generalized eigenvector | $\mathbf w=(1,-1,0)$, solving $(A-I)\mathbf w=\mathbf v_2$ |
| Structure | one genuine $2\times2$ Jordan block at $\lambda=1$ |
Additional Notes
- The secular term $C_3(t+1)e^t$ etc.\ is the hallmark of a Jordan block: it grows algebraically faster than a pure exponential mode.
- Verified by direct substitution into all three original equations, matching coefficients of $e^{2t}$, $e^t$, and $te^t$.
Result
The general solution of the system is: