Consider a system of two toy railway cars (i.e., frictionless masses) connected to each other by two springs, one of which is attached to the wall, as shown in the figure. Let $x_1$ and $x_2$ be the displacement of the first and second masses from their equilibrium positions. Suppose the masses are $m_1 = 6 \ \mathrm{kg}$ and $m_2 = 3 \ \mathrm{kg}$, and the spring constants are $k_1 = 48 \ \mathrm{N/m}$ and $k_2 = 24 \ \mathrm{N/m}$.

1. Set up a system of second-order differential equations that models this situation.

   $\boldsymbol{\vec{x}^{\, \prime\prime}} =$

   $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right[$ $\left.\vphantom{\begin{array}{c}\!\strut\\\!\strut\\\!\strut\\\end{array}}\right]$

   $\boldsymbol{\vec{x}}$

   
   
   
2. Find the general solution to this system of differential equations. Use $a_1, a_2, b_1, b_2$ to denote arbitrary constants, and enter them as a1, a2, b1, b2.
   $x_1(t) =$
   $x_2(t) =$

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