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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! 

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### 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.

Convert the second-order system of ODEs describing the two-mass/three-spring problem into a linear system of four first-order ODEs.

Learn how to transform a two-mass/three-spring second-order ODE system into a four first-order ODE system. This step-by-step solution involves defining variables, substituting given constants, and writing a linear system suitable for engineering and physics students.

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