The problem involves converting the given second-order system of ordinary differential equations (ODEs) into a system of four first-order ODEs. Let’s break it down step-by-step.

The second-order equations are: $m_1 \frac{d^2x_1}{dt^2} = -(k_1 + k_2)x_1 + k_2x_2,$ $m_2 \frac{d^2x_2}{dt^2} = k_2x_1 - (k_2 + k_3)x_2.$

The values provided are: $m_1 = 1, \quad m_2 = \frac{3}{2}, \quad k_1 = \frac{3}{2}, \quad k_2 = 2, \quad k_3 = 1.$

To convert this into a first-order system, we define the variables: $u_1 = x_1, \quad u_2 = x_2, \quad u_3 = \frac{dx_1}{dt}, \quad u_4 = \frac{dx_2}{dt}.$

Step 1: Substituting variables

Rewrite the original equations using the new variables. For the second derivatives: $\frac{d^2x_1}{dt^2} = \frac{du_3}{dt}, \quad \frac{d^2x_2}{dt^2} = \frac{du_4}{dt}.$

Using the first-order variables, we form the following system:

1. $\frac{du_1}{dt} = u_3$,
2. $\frac{du_2}{dt} = u_4$,
3. From the first equation: $m_1 \frac{du_3}{dt} = -(k_1 + k_2)u_1 + k_2u_2,$ which simplifies to: $\frac{du_3}{dt} = \frac{-(k_1 + k_2)}{m_1}u_1 + \frac{k_2}{m_1}u_2.$
4. From the second equation: $m_2 \frac{du_4}{dt} = k_2u_1 - (k_2 + k_3)u_2,$ which simplifies to: $\frac{du_4}{dt} = \frac{k_2}{m_2}u_1 - \frac{(k_2 + k_3)}{m_2}u_2.$

Step 2: Substituting parameter values

Substitute $m_1 = 1$, $m_2 = \frac{3}{2}$, $k_1 = \frac{3}{2}$, $k_2 = 2$, $k_3 = 1$ into the system:

1. $\frac{du_1}{dt} = u_3$,
2. $\frac{du_2}{dt} = u_4$,
3. $\frac{du_3}{dt} = -\left(\frac{\frac{3}{2} + 2}{1}\right)u_1 + \frac{2}{1}u_2 = -\frac{7}{2}u_1 + 2u_2$,
4. $\frac{du_4}{dt} = \frac{2}{\frac{3}{2}}u_1 - \frac{(2 + 1)}{\frac{3}{2}}u_2 = \frac{4}{3}u_1 - 2u_2$.

Step 3: Final system

The system of four first-order ODEs is: $\frac{du_1}{dt} = u_3,$ $\frac{du_2}{dt} = u_4,$ $\frac{du_3}{dt} = -\frac{7}{2}u_1 + 2u_2,$ $\frac{du_4}{dt} = \frac{4}{3}u_1 - 2u_2.$

Let me know if you’d like further clarifications or help visualizing this!

Questions for Further Exploration:

1. How can the system matrix $A$ for $\mathbf{u}' = A\mathbf{u}$ be derived from the above system?
2. What are the eigenvalues of the matrix $A$, and what do they signify physically?
3. How would the system change if external forcing terms $F_1(t)$ and $F_2(t)$ were included?
4. Can this system be solved explicitly, and if so, what are the solutions?
5. How does changing one of the spring constants (e.g., $k_3$) affect the system's dynamics?

Tip:

When converting second-order systems to first-order, always clearly define your substitution variables to avoid confusion in subsequent steps.