Just from looking at the slope field (image attached below) for the differential equation $\displaystyle\frac{dy}{dt}=f(t,y)$, I would say that $y=-1+e^{-t}$ is a possible solution. This is, however, apparently incorrect. I can't see why it would be incorrect, and can't really see a solution in a form other than that of $y=-1+Ce^{-t}$ for real $C$ values.
What could a possible solution be?
EDIT: The vertical axis is the $y$ axis, and the horizontal axis is the $t$ axis. Neither the function $f(t,y)$ nor the way in which the slopes were drawn are given.
$e^{-t}$ isn't the only function that tends to $0$. Looking at the slope field, there may or may not be a moving vertical asymptote as well. In that case, a possible solution might be
$$ y(t) = -1 + \frac{1}{(x-c)^n} $$
For some $n \ge 0$