I'm trying to think of a function $x(t)$ such that $\dot{x}(t) = x(t)^2t.$ I worked out for my self that it has something to do with the $x(t) = -2t^{-2}$ but I couldn't work out how to put the constant in so that the function still worked.
Since then I've been told that the function is $x(t) = \frac{2}{c-t^2}$ and I can easily see that this function works but I can't work out how to get to that answer without just having to spot it. Is there a way to work this out or does one just need to be good at spotting these things?