Consider a mechanical system shown in figure. Masses are free to slide over frictionless horizontal surface. The equation of motion of mass $$m_1$$ is

$$m_1\ddot{x}_1 + (\lambda_1 + \lambda_2)\dot{x}_1 + \lambda_2 \dot{x}_2 - (k_1 + k_2)x_1 - k_2x_2 = F_2$$

$$m_2\ddot{x}_1 + (\lambda_1 - \lambda_2)\dot{x}_1 + \lambda_2 \dot{x}_2 - (k_1 - k_2)x_1 - k_2x_2 = F_1$$

$$m_1\ddot{x}_1 + (\lambda_1 + \lambda_2)\dot{x}_1 - \lambda_2 \dot{x}_2 + (k_1 + k_2)x_1 - k_2x_2 = F_1$$

$$m_1\ddot{x}_1 + (\lambda_1 - \lambda_2)\dot{x}_1 + \lambda_2 \dot{x}_2 - (k_1 - k_2)x_1 - k_2x_2 = F_2$$