I was reading a book of ODE and make a comment using the uniqueness theorem on the behavior of the solutions of an autonomous differential equation $ \frac{dy}{dt} = f \left(y \right) $ Suppose $f$ satisfies the hypotheses of the theorem of existence and uniqueness. Take for example $ f\left( y \right) = \left( {y - 2} \right)\left( {y + 1} \right) $
Then we know from the theorem that the solutions do not intersect (the uniqueness). Suppose you have a solution satisfying $ y (0) = 1 / 2 $. Clearly $y(t)$ is between -1 and 2, is also decreasing, and bounded, so must tend asymptotically to a horizontal line, which should be $y =c$ for some $c\ge-1$. Why actually happens it tends to the line of phase? That is, why do we actually get $c=-1$? Do you know any good books where you can learn these things? an introductory book?