Doubt in the solution of differential equation 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?

It is a classical differential equation called the logistic differential equation http://math.usu.edu/~powell/biomath/mlab3-02/node2.html
