Q: Interesting change of variable for a "simple" ODE I'm working on a dynamical system governed by the following ODE in $\mathbb{R}^n$:
$$ \dot{x}(t) = f(A(t)x(t)+b). $$
Here $t\rightarrow A(t)\in\mathbb{R}^{n\times n}$ is a smooth (enough) matrix valued function and $b\in\mathbb{R}^n$ and $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ is smooth (enough) too. So in general the dynamics is just given by the composition between a non-linear transformation and an affine one.
My question is if there is some famous/interesting change of variable which allows to rewrite this system in some useful way, i.e. to Hamiltonian system or gradient vector field or something more manageable. You can even tell me that something is a good choice when $A$ satisfies some particular property, it would still be very helpful.
I see it is a quite general question and I am not saying what are my aims but, at the moment, I just want to see if in this setting there is some "standard" way to proceed.
The easiest transformation, i.e. $z(t)=A(t)x(t)+b$, does not seem to bring anywhere:
$$ \dot{z}(t) = \dot{A}(t)A^{-1}(t)(z(t)-b)+A(t)f(z(t)),$$
assuming the invertibility of $A$.

A: As
$$
\dot z = \dot A x + A \dot  x
$$
assuming $A$ invertible
$$
\dot x = A^{-1}(\dot z - \dot A x) = A^{-1}(\dot z - \dot A A^{-1}(z-b))
$$
then
$$
A^{-1}(\dot z - \dot A A^{-1}(z-b)) = f(z)
$$
or
$$
\dot z = A f(z)+\dot A A^{-1}(z-b)
$$
