The Lagrangian of two coupled oscillators of mass m each is $$L = \frac{1}{2}\left( {{{\dot x}_1}^2 + {{\dot x}_2}^2} \right) - \frac{1}{2}m{\omega _0}^2\left( {{x_1}^2 + {x_2}^2} \right) + m{\omega _0}^2\mu {x_1}{x_2}$$
The equations of motion are

A. $${{\ddot x}_1} + {\omega _0}^2{x_1} = {\omega _0}^2\mu {x_1},\,{{\ddot x}_2} + {\omega _0}^2{x_2} = {\omega _0}^2\mu {x_2}$$

B. $${{\ddot x}_1} + {\omega _0}^2{x_1} = {\omega _0}^2\mu {x_2},\,{{\ddot x}_2} + {\omega _0}^2{x_2} = {\omega _0}^2\mu {x_1}$$

C. $${{\ddot x}_1} + {\omega _0}^2{x_1} = {\omega _0}^2\mu {x_1},\,{{\ddot x}_2} + {\omega _0}^2{x_2} = - {\omega _0}^2\mu {x_2}$$

D. $${{\ddot x}_1} + {\omega _0}^2{x_1} = {\omega _0}^2\mu {x_1},\,{{\ddot x}_2} + {\omega _0}^2{x_2} = {\omega _0}^2\mu {x_1}$$

Answer: Option B