Solve the system of differential equations: dot x=3x-2y-z…
Solve the system of differential equations: $$ \dot x=3x-2y-z,\qquad \dot y=3x-4y-3z,\qquad \dot z=2x-4y $$
Explanation of Method
The characteristic cubic factors as $-(\lambda-2)^2(\lambda+5)$; since the eigenspace at $\lambda=2$ turns out to be a full plane, the matrix is diagonalisable and the solution is a straightforward superposition of three exponential modes.
- Write the system in matrix form and compute the characteristic polynomial.
- Identify the repeated eigenvalue $\lambda=2$ and the simple eigenvalue $\lambda=-5$.
- Find the eigenspace at $\lambda=2$ and the eigenvector at $\lambda=-5$.
- Assemble and verify the general solution.
Step 1 · Matrix form and characteristic equation
With $A=\begin{pmatrix}3&-2&-1\\3&-4&-3\\2&-4&0\end{pmatrix}$, expanding $\det(A-\lambda I)$ gives
$$ -\lambda^3-\lambda^2+16\lambda-20=-(\lambda-2)^2(\lambda+5)=0. $$
Eigenvalues: $\lambda_1=2$ (algebraic multiplicity 2) and $\lambda_2=-5$ (simple).
Step 2 · Eigenvector for $\lambda_2=-5$
$$ \mathbf v_1=\begin{pmatrix}1\\3\\2\end{pmatrix},\qquad A\mathbf v_1=-5\mathbf v_1.\ \checkmark $$
Step 3 · Eigenspace for $\lambda_1=2$
$A-2I$ has rank 1 (all rows proportional to $(1,-2,-1)$), giving $v_1=2v_2+v_3$ with two free parameters — geometric multiplicity 2, equal to algebraic multiplicity. Two independent eigenvectors:
$$ \mathbf p=\begin{pmatrix}2\\1\\0\end{pmatrix},\qquad \mathbf q=\begin{pmatrix}1\\0\\1\end{pmatrix}. $$
$A$ is diagonalisable at $\lambda=2$ — no secular ($te^{2t}$) term is required.
Step 4 · General solution and verification
The three eigenvectors are independent ($\det=-7\neq0$); substitution confirms the general solution below satisfies all three original equations. ✓
Summary Table
| Quantity | Value |
|---|---|
| Eigenvalues | $\lambda=2$ (mult.\ 2), $\lambda=-5$ (simple) |
| Eigenspace at $\lambda=2$ | the plane $v_1=2v_2+v_3$ |
| Structure | diagonalisable, no Jordan blocks |
Additional Notes
- As $t\to\infty$ the solution is dominated by the growing $e^{2t}$ modes; the $e^{-5t}$ term decays away.
- Verified by direct substitution into all three original equations.
Result
The general solution of the system is: