Chapter 6.2, Problem 33E

On a smooth horizontal surface, a mass of $m_1$ kg is attached to a fixed wall by a spring with spring constant $k_{1}N/m$. Another mass of $m_2$ kg is attached to the first object by a spring with spring constant $k_{2}N/m$. The objects are aligned horizontally so that the springs are their natural lengths. As we showed in Section $5.6$, this coupled mass-spring system is governed by the system of differential equations

$m_1\frac{d^{2}x}{d{t}^2}+\left(k_1+k_2\right)x-k_{2}y=0$, $\left(34\right)$

$m_2\frac{d^{2}y}{d{t}^2}-k_{2}x+k_{2}y=0$. $\left(35\right)$

Let’s assume that $m_1=m_2=1$, $k_1=3$, and $k_2=2$. If both objects are displaced $1m$ to the right of their equilibrium positions (compare Figure $5.6$, page $283$) and then released, determine the equations of motion for the objects as follows:

a. Show that $x\left(t\right)$ satisfies the equation

$x^{\left(4\right)}\left(t\right)+7x^{″}\left(t\right)+6x\left(t\right)=0$. $\left(36\right)$

b. Find the general solution $x\left(t\right)$ to $\left(36\right)$.

c. Substitute $x\left(t\right)$ back into $\left(34\right)$ to obtain a general solution for $y\left(t\right)$.

d. Use the initial conditions to determine the solutions, $x\left(t\right)$ and $y\left(t\right)$, which are the equations of motion.