5
You want a numerical solution, but this might help you check your computed results.
If $a$ satisfies the ODE, you know $e^{a(t)}a'(t) = f(t)$. Integrating you get
\begin{align}
\int_0^t\, f(\tau)\, d\tau &= \int_0^t e^{a(\tau)}a'(\tau)\, d\tau \\
&= \int_{a(0)}^{a(t)} e^a da \\
&= e^{a(t)} - e^{a(0)}.
\end{align}
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