Answer
It is not linear.
Work Step by Step
The logistic differential equation is of the form
$$ \frac{d y}{d t}=k y\left(1-\frac{y}{A}\right) =ky-\frac{ky^2}{A}$$
which is not of the form of a linear differential equation (notice the $y^2$ term). Thus, it is not linear.
