Step 3: Replacing derivatives by finite differences

Inserting the finite difference approximation in $$u'(t_n) = -au(t_n)$$ gives $$\begin{equation} \frac{u^{n+1}-u^{n}}{t_{n+1}-t_n} = -au^{n},\quad n=0,1,\ldots,N_t-1 \tag{5} \end{equation}$$

(Known as discrete equation, or discrete problem, or finite difference method/scheme)