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

$x_1(0)=0$

,

$x^{\prime }1(0)=-1$

,

$x_2(0)=0$

, and

$x{\text{'}}_2(0)=1$

.