Model problem and numerical solution method

Problem: $$\begin{equation} u'(t) = -au(t),\quad u(0)=I,\ 0 < t \leq T, \tag{3} \end{equation}$$

Solution method ($\theta$-rule): $$u^{n+1} = \frac{1 - (1-\theta) a\Delta t}{1 + \theta a\Delta t}u^n, \quad u^0=I\tp$$