$\huge \begin{align} \Delta x &= (x_{2}-x_{1}) -L \\ E_{m_{1}s} &= +\Delta x k \\ E_{m_{2}s} &= -\Delta x k \\ \end{align} $ $\ha{ m_{1} \ddot x_{1}(t) - k \Delta x &= 0\\ -m_{1} m_{2} \ddot x_{1}(t) + km_{2} \Delta x &= 0 \\ \\ m_{2} \ddot x_{2}(t) + k \Delta x &= 0\\ -m_{1} m_{2} \ddot x_{2}(t) + km_{m} \Delta x &= 0 \\ \\ m_{1}m_{2}( \ddot x_{2} -\ddot x_{1}) + (m_{1}+m_{2})k\Delta x &= 0\\ }$ $ \huge \begin{align} \mathcal D^2\{ x_{2} - x_{1} \} &= \mathcal D^2 (\Delta x - L) \\ 0 &= \mathcal D^2\left\{ \Delta x \right\} - \mathcal D^2\left\{ L \right\} \\ m_{1}m_{2} \mathcal D^2\{\Delta x\} + (m_{1}+m_{2})k \Delta x &= 0 \\ \mathcal D^2 \Delta x + \frac{k}{ \frac{m_{1}m_{2}}{m_{1}+m_{2}} } \Delta x &= 0 \\ M_{r} &= \frac{m_{1}m_{2}}{m_{1}+m_{2}} \\ \end{align}$