Hello I haven't seen a question like this before and would appreciate any help or guidance with the question:
Let $x, y : I \to \Bbb R^N$ , where $N \in \Bbb N$, be solutions of
$$x'(t) = f(t)$$
$$x(t_0) = x_0$$
and
$$y′(t) = f(t)$$
$$y(t_0) = y_0$$
where $t_0 \in I$, $I ⊂ \Bbb R$ is an open interval, $x_0, y_0 \in \Bbb R^N$ and $f : I \to \Bbb R^N$ is a continuous function.
Show that
$$\Vert x(t) − y(t) \Vert = \Vert x_0 − y_0 \Vert$$
holds for all $ \in I$.
I can solve it when x'(t) is equal to an equation but Im not sure what to do when its just a continuous function or how to solve it in this case. Im also not sure how xo and yo can be linked together.
Any help is appreciated thanks a lot.
$x(t)=\int_{t_0}^t f(s) ds +x_0$ and $y(t)=\int_{t_0}^t f(s) ds +y_0$
hence, $|| x(t) - y(t) || = ||x_0 - y_0||$