$$V(x_0, x_1)=\frac{1}{2}K(x_0-x_1)^2$$

$$F_0=-\frac{\partial}{\partial x_0}V(x_0, x_1)=-K(x_0-x_1)$$

$$F_1=-\frac{\partial}{\partial x_1}V(x_0, x_1)=K(x_0-x_1)$$

$$m_0\frac{d^2}{dt^2}x_0(t)=F_0=-K(x_0(t)-x_1(t))$$

$$m_1\frac{d^2}{dt^2}x_1(t)=F_1=K(x_0(t)-x_1(t))$$

$$x_0(t) = X_0(\omega)e^{-i\omega t}$$

$$x_1(t) = X_1(\omega)e^{-i\omega t}$$

$$m_0\frac{d^2}{dt^2}x_0(t)=-m_0\omega^2X_0(\omega)e^{-i\omega t}=-K(X_0(\omega)e^{-i\omega t}-X_1(\omega)e^{-i\omega t}))$$

$$m_1\frac{d^2}{dt^2}x_1(t)=-m_1\omega^2X_1(\omega)e^{-i\omega t}=K(X_0(\omega)e^{-i\omega t}-X_1(\omega)e^{-i\omega t}))$$