Laplace Transforms: Vertical Motion of a Coupled Spring System

Problem Statement

Figure 1. Coupled Spring System.

Derive the system of differential equations describing the straight-line vertical motion of the coupled spring shown in Figure 1. Use Laplace transform to solve the system when $k_{1}=k_{2}=k_{3}=1$, $m_{1}=m_{2}=1$, and $x_{1}(0)=0$, $x'1(0)=-1$, $x_{2}(0)=0$, and $x'_{2}(0)=1$.