Ok, let $M$ be a smooth manifold and $X$ a smooth vector field on $M$. Then $V\colon=\varphi_{\ast}X\colon\mathbb{R}^n\rightarrow T\mathbb{R}^n,\,x\mapsto d\varphi_{\varphi^{-1}(x)}(X(\varphi^{-1}(x)))$ is a smooth vector field on $\mathbb{R}^n$. We identify $T_x\mathbb{R}^n\cong\mathbb{R}^n$ canonically for all $x\in\mathbb{R}^n$ and think of $V$ as a map $\mathbb{R}^n\rightarrow\mathbb{R}^n$.
For each $x\in\mathbb{R}^n$, the IVP $y^{\prime}=V(y),\,y(0)=x$ has a unique solution $f_x\colon(-\varepsilon_x,\varepsilon_x)\rightarrow\mathbb{R}^n$ for some $\varepsilon_x>0$. Now we want these to…