The vector field $ \frac{d}{dx} $ over the manifold $ \mathbb R $ is a complete vector field as the solution of the integral equations $$ \frac{dx(t)}{dt} = 1; $$ Has the solution, $x(t) = t + x(0)$ , so the solution can be extended for all $t \in \mathbb R $,
Now, if we remove one single point from $\mathbb R$, i.e $\mathbb R \setminus \{0\}$, then in finite time the solution goes off the manifold at $t=0$, That's why in the modified manifold the vector field fails to be complete.
Now, If I modify the manifold to be $\mathbb R_+$, Then also in finite time the solution flies off the manifold, $t \leq 0$, So here also the vector field fails to be a complete vector field.
Now, along the same chain of thought I don't understand why the vector field $x \frac{d}{dx}$ in $\mathbb R_+$ is a complete vector field, The solution of the integral curve is $$ \frac{dx(t)}{dt} = x(t),\quad x(t) = x(0)e^t $$ For any $x(0) \in \mathbb R_+ $, the solution goes off $\mathbb R_+$ in finite time, So it should not be called a complete vector?
What I'm doing wrong here?