Problem Statement
A mechanical damper system is governed by the coupled linear differential system:
$$\begin{pmatrix} \dot{x} \\ \dot{y} \end{pmatrix} = \begin{pmatrix} -1 & 2 \\ -3 & -6 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}$$
**(a)** [4 marks] Find the eigenvalues and corresponding eigenvectors of the coefficient matrix.
**(b)** [4 marks] Classify the nature and stability of the equilibrium point at $(0,0)$ and write down the general solution for $x(t)$ and $y(t)$.
Verified Solution & Marking Scheme
Part (a) [M1]
Characteristic equation: $\det(A - \lambda I) = (-1 - \lambda)(-6 - \lambda) - (2)(-3) = \lambda^2 + 7\lambda + 6 + 6 = \lambda^2 + 7\lambda + 12 = 0$
Part (a) [A1]
Factor: $(\lambda + 3)(\lambda + 4) = 0 \implies \lambda_1 = -3, \lambda_2 = -4$
Part (a) [M1]
For $\lambda = -3$: $\begin{pmatrix} 2 & 2 \\ -3 & -3 \end{pmatrix}\begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \mathbf{0} \implies v_1 + v_2 = 0 \implies \mathbf{v}_1 = \begin{pmatrix} 1 \\ -1 \end{pmatrix}$
Part (a) [A1]
For $\lambda = -4$: $\begin{pmatrix} 3 & 2 \\ -3 & -2 \end{pmatrix}\begin{pmatrix} v_1 \\ v_2 \end{pmatrix} = \mathbf{0} \implies 3v_1 + 2v_2 = 0 \implies \mathbf{v}_2 = \begin{pmatrix} 2 \\ -3 \end{pmatrix}$
Part (b) [R1]
Since both eigenvalues are real, distinct, and strictly negative ($\lambda_1 = -3 < 0, \lambda_2 = -4 < 0$), the equilibrium point $(0,0)$ is a **stable node (sink)**
Part (b) [A1]
All trajectories converge asymptotically to $(0,0)$ as $t \to \infty$
Part (b) [A1]
General solution vector: $\begin{pmatrix} x(t) \\ y(t) \end{pmatrix} = c_1 e^{-3t} \begin{pmatrix} 1 \\ -1 \end{pmatrix} + c_2 e^{-4t} \begin{pmatrix} 2 \\ -3 \end{pmatrix}$