Let $f \in C ^ 1 (\mathbb{R^n})$ for each $x_0 \in \mathbb{R^n}$, the initial value problem
$$ x'=\frac{f(x)}{1+ f(x)} $$
$$ x(0)=x_0\tag{2} $$
has a unique solution $x(t)$ defined for all $t \in \mathbb{R^n}$
This, in fact, I already managed to prove!
I need to prove that the sistem (2) is topologically equivalent to the system $x '= f (x)$ (1) , through the homeomorphism $H = Id_\mathbb{R ^ n}$
since the time t along the solutions $x(t)$ of (1) has simply been rescaled according to the formula
$$ \tau = \int\limits_0^t 1+ ||f(x(s)||\space ds $$
I know that $\large{\frac{dx}{d\tau}=\frac{dx}{dt}/\frac{d\tau}{dt}=\frac{f(x)}{1+ ||f(x)||}}$
I need to prove that $H = Id_\mathbb{R ^ n}$