Truncation error of the $\theta$-rule

The $\theta$-rule: $$[\bar D_t u = -a\overline{u}^{t,\theta}]^{n+\theta} \tp$$

Truncation error: $$[\bar D_t \uex + a\overline{\uex}^{t,\theta} = R]^{n+\theta} \tp$$ Use (11)-(12) and (17)-(18) along with $\uex'(t_{n+\theta}) + a\uex(t_{n+\theta})=0$ to show

$$\begin{align} R^{n+\theta} = &({\half}-\theta)\uex''(t_{n+\theta})\Delta t + \half\theta (1-\theta)\uex''(t_{n+\theta})\Delta t^2 + \nonumber\\ & \half(\theta^2 -\theta + 3)\uex'''(t_{n+\theta})\Delta t^2 + \Oof{\Delta t^3} \end{align}$$ Note: 2nd-order scheme if and only if $\theta =1/2$.