Question: I know that the solution to the differential equation $\frac{dy}{dt}=y(a-by), a>0, b>0, y(0)=y\_0$ can be derived using integration using partial fractions, and the final result is: $$y=a/(b+ke^{-at})$$ where $k$ is $(a/y\_0)-b$.

My doubt is: Say $y\_0<0$. Then the differential equation itself tells me that $\frac{dy}{dt}<0$ and hence $y$ would just go on falling as $t$ increases, and it would always remain negative. But if you look at the behaviour of the function $y$, there would be a vertical asymptote at some positive value of $t$ (to be precise, where $b+ke^{-at}=0$. Beyond this value of $t$, $y$ would be positive. What am I getting wrong here?

Answer: 
You got nothing wrong, it is just that the solution ends at the asymptote. Solutions of differential equations are defined over intervals containing the initial point. These intervals can have finite end points even for the so-called "maximal solution". Often these finite end points happen because the trajectory of the solution moves to infinity in finite time, as happens here.
It is a classical differential equation called the **logistic differential equation** <http://math.usu.edu/~powell/biomath/mlab3-02/node2.html>