Example of differential equation

Question

Can you give an example of differential equation $F(t, x, x')$ which is defined everywhere on $\mathbb{R}^3$, but there is no solution which is defined everywhere on $\mathbb{R}$. I tried to think about the answer, however, all my examples has solution $x = 0$ or something like this, which violates condition, because such functions are defined everewhere on $\mathbb{R}$.