To calculate \( x_1(t) \) and \( x_2(t) \) for the given coupled spring-mass system, we need to solve the system of differential equations obtained from the Lagrange equations. These equations are:

\[
m_1 \ddot{x}_1 + (k_1 + k_2) x_1 - k_2 x_2 = 0
\]
\[
m_2 \ddot{x}_2 + (k_2 + k_3) x_2 - k_2 x_1 = 0
\]

### Step 1: Set up the System in Matrix Form

We can write the system of differential equations in matrix form as:

\[
\begin{pmatrix}
m_1 & 0 \\
0 & m_2
\end{pmatrix}
\begin{pmatrix}
\ddot{x}_1 \\
\ddot{x}_2
\end{pmatrix}
+
\begin{pmatrix}
k_1 + k_2 & -k_2 \\
-k_2 & k_2 + k_3
\end{pmatrix}
\begin{pmatrix}
x_1 \\
x_2
\end{pmatrix}
=
\begin{pmatrix}
0 \\
0
\end{pmatrix}
\]

This can be expressed as:

\[
\mathbf{M} \ddot{\mathbf{x}} + \mathbf{K} \mathbf{x} = \mathbf{0}
\]

where:

\[
\mathbf{M} = \begin{pmatrix}
m_1 & 0 \\
0 & m_2
\end{pmatrix},
\quad
\mathbf{K} = \begin{pmatrix}
k_1 + k_2 & -k_2 \\
-k_2 & k_2 + k_3
\end{pmatrix},
\quad
\mathbf{x} = \begin{pmatrix}
x_1 \\
x_2
\end{pmatrix}
\]

### Step 2: Assume Solutions of the Form

Assume the solutions are harmonic:

\[
x_1(t) = A_1 \cos(\omega t) + B_1 \sin(\omega t)
\]
\[
x_2(t) = A_2 \cos(\omega t) + B_2 \sin(\omega t)
\]

Plugging these into the differential equations gives us:

\[
-m_1 \omega^2 A_1 + (k_1 + k_2) A_1 - k_2 A_2 = 0
\]
\[
-m_2 \omega^2 A_2 + (k_2 + k_3) A_2 - k_2 A_1 = 0
\]

This can be rewritten as:

\[
\left[ \mathbf{K} - \omega^2 \mathbf{M} \right] \mathbf{A} = \mathbf{0}
\]

where \( \mathbf{A} = \begin{pmatrix} A_1 \\ A_2 \end{pmatrix} \).

### Step 3: Solve the Characteristic Equation

The above system has a non-trivial solution if the determinant of the matrix \( \mathbf{K} - \omega^2 \mathbf{M} \) is zero:

\[
\det \left( \mathbf{K} - \omega^2 \mathbf{M} \right) = 0
\]

This determinant gives a quadratic equation in \( \omega^2 \):

\[
\left[ (k_1 + k_2) - m_1 \omega^2 \right] \left[ (k_2 + k_3) - m_2 \omega^2 \right] - k_2^2 = 0
\]

Expanding and simplifying, you can solve for \( \omega^2 \), which gives you the natural frequencies \( \omega_1 \) and \( \omega_2 \).

### Step 4: Compute \( A_1 \) and \( A_2 \)

For each natural frequency \( \omega_1 \) and \( \omega_2 \), you can substitute back into the equations to find the ratio \( \frac{A_1}{A_2} \).

### Step 5: General Solution

Finally, the general solution for the system is a combination of the two normal modes:

\[
x_1(t) = C_1 \cos(\omega_1 t + \phi_1) + C_2 \cos(\omega_2 t + \phi_2)
\]
\[
x_2(t) = D_1 \cos(\omega_1 t + \psi_1) + D_2 \cos(\omega_2 t + \psi_2)
\]

### Example Calculation

Let's assume we calculate specific values for \( \omega_1 \) and \( \omega_2 \), then \( x_1(t) \) and \( x_2(t) \) would depend on these frequencies, and the constants \( C_1, C_2, D_1, D_2 \) are determined by initial conditions.

If you provide the masses (\( m_1, m_2 \)) and spring constants (\( k_1, k_2, k_3 \)), I can perform the exact calculations for \( x_1(t) \) and \( x_2(t) \).

Would you like to proceed with a specific numerical example or have any other questions?

Learn how to calculate X1(t) and X2(t) for a coupled spring-mass system using Lagrange equations and harmonic motion principles. This problem involves differential equations, matrix operations, and harmonic motion, suitable for advanced undergraduate students and higher.

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