Solve the system of differential equations: dot x=-x+y-2z…
Solve the system of differential equations: $$ \dot x=-x+y-2z,\qquad \dot y=4x+y,\qquad \dot z=2x+y-z $$
Explanation of Method
The eigenvalue $\lambda=-1$ is repeated but has only a one-dimensional eigenspace, forcing a generalized eigenvector and a $te^{-t}$ secular term, while the simple eigenvalue $\lambda=1$ contributes a single growing exponential mode.
- Write the system in matrix form and compute the characteristic polynomial.
- Identify the simple eigenvalue $\lambda=1$ and the repeated eigenvalue $\lambda=-1$.
- Find the eigenvector at $\lambda=-1$ 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 $A=\begin{pmatrix}-1&1&-2\\4&1&0\\2&1&-1\end{pmatrix}$, expanding $\det(A-\lambda I)$ gives
$$ -\lambda^3-\lambda^2+\lambda+1=-(\lambda+1)^2(\lambda-1)=0. $$
Eigenvalues: $\lambda_1=-1$ (algebraic multiplicity 2) and $\lambda_2=1$ (simple).
Step 2 · Eigenvector for $\lambda_2=1$
$$ \mathbf v_1=\begin{pmatrix}0\\2\\1\end{pmatrix},\qquad A\mathbf v_1=\mathbf v_1.\ \checkmark $$
Step 3 · Eigenvector for $\lambda_1=-1$ and its rank deficiency
$A+I=\begin{pmatrix}0&1&-2\\4&2&0\\2&1&0\end{pmatrix}$ has rank 2, so the null space is one-dimensional:
$$ \mathbf v_2=\begin{pmatrix}-1\\2\\1\end{pmatrix},\qquad A\mathbf v_2=-\mathbf v_2.\ \checkmark $$
Geometric multiplicity $1<2$: a generalized eigenvector is required.
Step 4 · Generalized eigenvector and Jordan-chain solutions
Solving $(A+I)\mathbf w=\mathbf v_2$ gives $\mathbf w=(1,-1,0)$, verified directly. The three independent solutions are
$$ \mathbf u^{(1)}=e^{t}\mathbf v_1,\qquad \mathbf u^{(2)}=e^{-t}\mathbf v_2,\qquad \mathbf u^{(3)}=e^{-t}(t\,\mathbf v_2+\mathbf w). $$
Step 5 · General solution and verification
The three solutions are independent at $t=0$ ($\det=1\neq0$); direct substitution confirms the general solution below satisfies all three original equations. ✓
Summary Table
| Quantity | Value |
|---|---|
| Eigenvalues | $\lambda=-1$ (mult.\ 2, geometric mult.\ 1), $\lambda=1$ (simple) |
| Generalized eigenvector | $\mathbf w=(1,-1,0)$ |
| Structure | one $2\times2$ Jordan block at $\lambda=-1$ |
Additional Notes
- The secular term decays ($te^{-t}\to0$ as $t\to\infty$), so the long-time behaviour is dominated entirely by the $C_1 e^t$ mode.
- Verified by direct substitution into all three original equations.
Result
The general solution of the system is: