I need t find the maximal integral curve starting at point $p \in \mathbb{R}$ of a vector field $x \frac{d}{dx}$ on $\mathbb{R}$.
Now, I know that for finding the integral field, I need to solve $$\frac{dx}{dt} = x $$
Solving them will give these two equations :- $$\ln x = t + c_1 \space $$ So the integral curve would be $x(t) = \exp(t+ c_1)$
I don't get what to do with this to find the maximal integral curve. In general how do we find the maximal integral curves?