Verhulst model and Lipschitz dependancy

Question

I have a differential equation as follow which is Verhulst model:
$$I'(t) = \beta I(t)\left(1-\dfrac {I(t)}N \right)$$ So I wanted to see just if there is a solution to this equation and if it is unique on [0, T] with 0€R. I derived the equation and I got
$$\dfrac {\partial }{\partial I} \beta \left(I(t)-\dfrac {I^2(t)}N \right)= \beta \left(1-\dfrac {2I(t)}N \right)$$ which is continuous on the interval and therefore there is a solution and it is unique. (Correct me here if I am wrong).
Now what I want to do is to find out if the constant of Lipschitz depends on T? (I don't want to calculate it, I just want to find out if it depends on T)
P.S. I would like to know also if this equation is linear or non-linear? I think that it is linear.