Consider a differential equation $\dot{x} = v(t,x)$. This differential equation is equivalent to autonomous differential equation $\dot{y} = w(y)$, with $y = (\alpha,x)$ and $w(y) = (1,v(t,x))$. By rectification theorem, we are always guaranteed a unique solution in a neighbourhood.

In Arnold's ODE book - chapter 2, section 4. He defines transformation over the time interval and gives a corollary. Which I couldn't prove? This image has the definition and corollary.

Here $\Phi(t_0,x_0;t)$ is the unique solution for the intial value problem $\dot{x} = v(t,x)$ and with $x(t_0)= x_0$

  • $\begingroup$ A hint for 3): when you go directly from time $t_0$ to $t$ the result is the same as when you go first from $t_0$ to $s$ and then from $s$ to $t$ (look at Fig. 70). $\endgroup$ – user539887 Mar 7 at 7:41

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