Question #68607

Let $M = \begin{pmatrix} 2 & -1 & 1 \\ 1 & 2 & -1 \\ 3 & 1 & 2 \end{pmatrix}$ and $\vec{b} = \begin{pmatrix} 3 \\ 2 \\ 11 \end{pmatrix}$. Let $\begin{pmatrix} x_0 \\ y_0 \\ z_0 \end{pmatrix}$ be the unique solution to the system $M \vec{X} = \vec{b}$. Consider the system of differential equations $\frac{d\vec{v}}{dt} = A\vec{v}$, where $A = \begin{pmatrix} -1 & -3 \\ 3 & -1 \end{pmatrix}$ and $\vec{v}(t) = \begin{pmatrix} x(t) \\ y(t) \end{pmatrix}$. Given the initial condition $\vec{v}(0) = \begin{pmatrix} x_0 \\ y_0 \end{pmatrix}$, let $L$ be the total arc length of the curve traced by $\vec{v}(t)$ in the $xy$-plane as $t$ varies from $0$ to $\infty$. Find the value of $L^2$.