A problem from Arnold's book I was reading Arnold's ODE book, there is written as corollary that 
Consider the differential equation $\frac{dx}{dt}= v(t, x)$ with $t\in \mathbb{R}$ and $x \in\mathbb{R^n}$. Then there exists a neighborhood $V$ of the point $(t_0,x_0)$ in $U$ and a diffeomorphism $f : V \to W$ such that the equation is equivalent to $\frac{dy}{dt}= 0$
Its given in page 56, as a corollary of basic theorem given in page $48$.
Can someone please explain me how to prove it ?
 A: Rewrite: (Rename the center point $x_0^*$ and $x^*(t)$ the solution through $(t_0,x_0^*)$, so $x_0$ is unclaimed.)
First assume that such $f$ exists. Then its inverse map takes a constant vector $y$ and produces for each $t$ a value $x(t)$, $f^{-1}(t,y)=(t,x(t))$ and by construction $x(t)$ is a solution of the differential equation. 
And indeed, there is a well-known map that has exactly these properties, the flow $φ(t;t_0,x_0)$, which gives for any initial condition $(t_0,x_0)$ the solution to the IVP
$$
φ(t_0;t_0,x_0)=x_0\text{ and }\partial_tφ(t;t_0,x_0)=v(t,φ(t;t_0,x_0))
$$
Thus one may attempt to construct $f$ from $f^{-1}(t,y)=(t,φ(t;t_0,y))$
One interesting fact about the flow is its composition property
$$
φ(s;t,φ(t;t_0,x_0))=φ(s;t_0,x_0)
$$ 
which allows us to directly construct the inverse operator from
$$
φ(t_0;t,φ(t;t_0,x_0))=φ(t_0;t_0,x_0)=x_0\implies f(t,x)=(t,φ(t_0;t,x))
$$
in words, the trajectory through $(t,x)$ is identified and its value $x_0=x(t_0)$ is returned as the identifying constant of the trajectory.
By the assumptions of Picard-Lindelöf, which is the least that has to be required, the flow is Lipschitz. Thus $f$ has been constructed as a Lipschitz homeomorphism.

To establish $f$ as a diffeomorphism one now only has to show that the flow is continuously differentiable as a function in all of its variables. For that task key assumptions are missing in the question that probably carry over from the assumption of the theorem preceding the corollary.
One such assumption is that $v$ is continuously differentiable with derivative bounded by a Lipschitz constant $L$. Then the desired result is an application of the perturbation theory for fixed point equations.

The domain $U$ is then most naturally a tube of trajectories around the center trajectory $x^*$. Its extension $(a,b)$ in time has to be chosen in such a way that it is contained in all the time intervals $(a(t_0,x_0),b(t_0,x_0))$ of the maximal solutions to all the contained IVP trajectories. The range $V$ is then $(a,b)$ times the section of $U$ at $t_0$. 
