I just want to bounce this off of the smart people on MSE to make sure I understand what's going on when we discuss complete vector fields.
Consider the following field. $X = e^{-x} \frac{\partial}{\partial x} + \frac{\partial}{\partial y}$. We would like to determine if this vector field is complete. To do so, we solve the system of ODEs defined by it, and check to see if all the integral curves are defined for all $t$.
The system of ODEs is $\frac{dx}{dt} = e^{-x}$, and $\frac{dy}{dt} = 1$. Solving the second equation first it is easy to observe that $y=t+k$. On the other hand, the first equation implies that $e^x dx = dt$ and hence that $e^x = t +c$, or that $x = \ln (t+c)$. But this is not defined for some $t$.
Am I correct in interpreting this to say that the integral curves can be thought of as tracing out the paths $e^x +c$ (of course the rate at which these curves are traced out goes down with $t)$? I believe that this vector field is not complete because this notion does not make sense for negative $t$ as we observed before. If I'm completely off base, how should I think about completness of vector fields, and how to check it?
edit: corrected some typos as pointed out in comments/answers.
Your reasoning is correct, except for the following: $e^x = t+c$ implies $x = \ln(t+c)$, not $\ln t+c$. The conclusion remains: the field is not complete.
More precisely, trajectories exist moving forward in time, i.e., if you start at some point $(x_0,y_0)$ at time $t=0$, then $c=e^{x_0^2+y_0^2}>0$, so the solution exists for $t>-e^{x_0^2+y_0^2}$.
The reason solutions explode when moving backward in time is that $x$ has to decrease as $t$ decreases, which makes $e^{-x}$ larger, which speeds up the process, etc — the feedback loop destroys the solution. Same thing happens with $x'=e^x$ moving forward in time, or with $x'=x^3$ going in either direction.