Using (15)-(16) and (5)-(6) we get \begin{align*} \lbrack mD_tD_t \uex + \beta D_{2t} \uex\rbrack^n &= m\uex''(t_n) + \beta\uex'(t_n) + \\ &\quad \left(\frac{m}{12}\uex''''(t_n) + \frac{\beta}{6}\uex'''(t_n)\right)\Delta t^2 + \Oof{\Delta t^4} \end{align*} The terms m\uex''(t_n) + \beta\uex'(t_n) + \omega^2\uex(t_n) + s(\uex(t_n)) - F^n, correspond to the ODE (= zero).
Result: accuracy of \Oof{\Delta t^2} since \begin{equation} R^n = \left(\frac{m}{12}\uex''''(t_n) + \frac{\beta}{6}\uex'''(t_n)\right)\Delta t^2 + \Oof{\Delta t^4}, \tag{39} \end{equation}
Correction terms: complicated when the ODE has many terms...