# Transformation over time interval of a non-autonomous differential equation

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 $$x(t_0)= x_0$$

• 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). – user539887 Mar 7 at 7:41