For the Logistic model it is denoted: \begin{equation} \begin{cases} \frac{dV(t)}{dt}=\alpha V(t)-\beta V(t)^2\\ V(0)=v_0 \end{cases} \end{equation} Now I need to find the exact solution?
I get to the stage where:
$\frac{-log(\alpha-\beta V(t))}{\alpha}+\frac{V(t)}{\alpha}=t+c_1$
However, how do I go about finding the exact solution?
Hint
Looking at what you wrote, I suppose that you rewrote $$V'=\alpha V - \beta V^2$$as $$\frac 1{t'}=\alpha V - \beta V^2\implies t'= \frac 1{\alpha V - \beta V^2}=\frac{1}{\alpha V}-\frac{\beta }{\alpha (\beta V-\alpha )}$$ Now, integrate properly both sides, isolate $V$ and apply the condition.