I was reading some texts online and i found this one intriguing:
Let $u$ be a real $C^1$ function defined on $[0,\infty)$ such that $ \partial_x u <u-\frac{2}{u}-1$ for every $t$. Find a $C>0$ such that $u>C$ everywhere.
If there wasn't that $\frac{2}{u}$, I would multiply by $e^{-t}$ then integrate to find the right inequality.Any idea?