$$ V(x_0, x_1)=\frac{1}{2}K(x_0-x_1)^2 $$
$$ F_0=-\frac{\partial}{\partial x_0}V(x_0, x_1)=-K(x_0-x_1) $$
$$ F_1=-\frac{\partial}{\partial x_1}V(x_0, x_1)=K(x_0-x_1) $$
$$ \omega_0=\sqrt\frac{K}{m_0} $$
$$ \omega_1=\sqrt\frac{K}{m_1} $$
$$ K=m_0\omega_0^2=m_1\omega_1^2 $$
$$ m_0\frac{d^2}{dt^2}x_0(t)=F_0=-K(x_0(t)-x_1(t)) $$
$$ m_1\frac{d^2}{dt^2}x_1(t)=F_1=K(x_0(t)-x_1(t)) $$
$$ x_0(t) = X_0(\omega)e^{-i\omega t} $$
$$ x_1(t) = X_1(\omega)e^{-i\omega t} $$
$$ m_0\frac{d^2}{dt^2}x_0(t)=-m_0\omega^2X_0(\omega)e^{-i\omega t}=-K(X_0(\omega)e^{-i\omega t}-X_1(\omega)e^{-i\omega t})) $$
$$ m_1\frac{d^2}{dt^2}x_1(t)=-m_1\omega^2X_1(\omega)e^{-i\omega t}=K(X_0(\omega)e^{-i\omega t}-X_1(\omega)e^{-i\omega t})) $$
$$ -m_0\omega^2X_0(\omega)=-KX_0(\omega)+KX_1(\omega) $$
$$ -m_1\omega^2X_1(\omega)=KX_0(\omega)-KX_1(\omega) $$
$$ (K-m_0\omega^2)X_0(\omega)-KX_1(\omega)=0 $$
$$ -KX_0(\omega)+(K-m_1\omega^2)X_1(\omega)=0 $$
$$ \begin{bmatrix} K-m_0\omega^2 & -K\\ -K & K-m_1\omega^2 \end{bmatrix} \begin{bmatrix} X_0(\omega)\\ X_1(\omega) \end{bmatrix}=O $$