Second-order systems and applications

Find the general solution to $\left[ \begin{smallmatrix} 1 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 3 \end{smallmatrix}\right] \vec{x}\,'' = \left[ \begin{smallmatrix} -3 & 0 & 0 \\ 2 & -4 & 0 \\ 0 & 6 & -3 \end{smallmatrix}\right] \vec{x} + \left[ \begin{smallmatrix} \cos(2t) \\ 0 \\ 0 \end{smallmatrix}\right]$.

Suppose there are three carts of equal mass $m$ and connected by two springs of constant $k$ (and no connections to walls). Set up the system and find its general solution.

Suppose a cart of mass 2 kg is attached by a spring of constant $k=1$ to a cart of mass 3 kg, which is attached to the wall by a spring also of constant $k=1$. Suppose that the initial position of the first cart is 1 meter in the positive direction from the rest position, and the second cart starts at the rest position. The carts are not moving and are let go. Find the position of the second cart as a f…