How to find a differential equation with exponential decay between two values

Question

Given the following the function which describes an exponential decay from A to $\alpha$*A in time with a time constant $\tau$ with $\alpha < 1$:

$$A+(\alpha*A-A)*(1-e^{(-t/\tau)}).$$

How can I find the associated differential equation with initial value A(0) = A. I've added a plot of the function below.

Thanks for any suggestions!

Ties

Answer 1

This is of the form $f(t) = u+ve^{wt}$.

Differentiating, $f'(t) =wve^{wt}$.

Now relate $f$ and $f'$.