= Solution
An <integral curve of a vector field> $X$ through $p$ is a <smooth curve> $c:I\to M$, on an interval containing zero, satisfying $c(0)=p$ and $\dot c(t)=X_{c(t)}$. The <local flow> is the smoothly varying family of these <integral curves of a vector field>:
$$
\phi:D\longrightarrow M,\qquad \phi_t(p)=c_p(t),\qquad
\phi_0(p)=p,\quad \partial_t\phi_t(p)=X_{\phi_t(p)}.
$$
Here $D$ is an open neighborhood of $\{0\}\times M$; for the maximal <local flow>, its fibre over $p$ is the maximal existence interval of $c_p$. Local existence, uniqueness and smooth dependence for a smooth <ordinary differential equation> give this family. Uniqueness also gives $\phi_{t+s}(p)=\phi_t(\phi_s(p))$ whenever the relevant curves and compositions are defined, and $\phi_{-t}=\phi_t^{-1}$ on their common domains.
Reparametrizing the <integral curves of a vector field> gives
$$
\frac{d}{dt}\phi_{2t}(p)=2X_{\phi_{2t}(p)}.
$$
Thus \b[the flow of $2X$ is $\boxed{\widetilde\phi_t=\phi_{2t}}$], with domain $\{(t,p):(2t,p)\in D\}$.
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