Solve the system of differential equations: dot x_1=x_2+x_3…
Solve the system of differential equations: $$ \dot x_1=x_2+x_3,\qquad \dot x_2=x_1+x_3,\qquad \dot x_3=x_1+x_2 $$
Explanation of Method
The coefficient matrix is symmetric with the same cyclic structure seen in earlier problems in this set; the repeated eigenvalue here has full geometric multiplicity, so two independent eigenvectors suffice without any generalised eigenvectors.
- Write the matrix $A$ and find the characteristic polynomial using row operations.
- Find the eigenvector for the simple eigenvalue.
- Find two independent eigenvectors for the repeated eigenvalue.
- Assemble the general solution.
Step 1 · Characteristic equation
With $A=\begin{pmatrix}0&1&1\\1&0&1\\1&1&0\end{pmatrix}$, adding all rows to the first and factoring gives
$$ (2-\lambda)(\lambda+1)^2=0, $$
so $\lambda_1=2$ (simple) and $\lambda_{2,3}=-1$ (double).
Step 2 · Eigenvector for $\lambda_1=2$
$$ \mathbf v_1=\begin{pmatrix}1\\1\\1\end{pmatrix}. $$
Step 3 · Eigenspace for $\lambda_{2,3}=-1$
Here $A+I$ has rank 1, so the eigenspace is genuinely two-dimensional (geometric multiplicity = algebraic multiplicity, no generalised eigenvectors needed). Choose
$$ \mathbf v_2=\begin{pmatrix}1\\-1\\0\end{pmatrix},\qquad \mathbf v_3=\begin{pmatrix}1\\0\\-1\end{pmatrix}. $$
Step 4 · Verification
Differentiating the general solution and substituting confirms all three original equations hold identically. ✓
Summary Table
| Quantity | Value |
|---|---|
| Eigenvalues | $\lambda_1=2$ (simple), $\lambda_{2,3}=-1$ (double, full geometric multiplicity) |
| Eigenvector for $\lambda_1$ | $(1,1,1)$ |
| Eigenspace for $\lambda_{2,3}$ | $\{v: v_1+v_2+v_3=0\}$ |
Additional Notes
- This is the same matrix structure as Problem 5 in this collection, and unsurprisingly gives the same eigenvalues and the same style of solution.
- Because $A+I$ has rank exactly 1 (all rows identical), the double eigenvalue is non-defective and no Jordan chain is needed.
- Verified by direct substitution into all three original equations.
Result
The general solution of the system is: