(I have posed this question over at math.se but since there were no answers I hope it's okay to post here.)

When dealing with (nonlinear) dynamical systems, one often deals with state space representation, i.e. systems of the form

$$\dot{x}=f(x),\quad x(t)\in\mathbb{R}^n.$$ Let $x^*$ be a solution of this system, then the variational equation reads

$$\dot{x}= \underbrace{\frac{\partial f}{\partial x}(x^*)}_{=:A}\cdot x\quad (1)$$

or with $D:=\frac{d}{dt}$ in operator form

$$\underbrace{(DI_n-A)}_{=:P(D)}x=0\quad(2)$$

where $I_n$ is the $n$-th unit matrix. The Elements of $P(D)$ in general are meromorphic functions in $D$, so we are dealing with Ore polynomial matrices with the shifting rule

$$Da=\dot{a}+aD$$ for all meromorphic functions $a$. Here is what I am very confused about: How can be assumed $\dot{x}=Dx$ when writing equation (1) in the form (2), and not $\dot{x}=Dx-xD$ (according to the shifting rule)? And why can't i factor out $D$ to the right instead, i.e. $\dot{x}=xD$ (which by the shifting rule would result in something different)?

Also, what I have encountered a lot, is implicit equations of the type $F(x,\dot{x})=0$ where the operator notation is acquired by $$0=\underbrace{\left(\frac{\partial F}{\partial \dot{x}}D+\frac{\partial F}{\partial x}\right)}_{=:P(D)}x.\quad (3)$$

Here too, I am wondering why this is not say $$0=\underbrace{\left(D\frac{\partial F}{\partial \dot{x}}+\frac{\partial F}{\partial x}\right)}_{=:P(D)}x\quad (4)$$

instead. What am I missing here?

**EDIT**: By $\dot{x}$ i mean the time derivative of $x$, so $\dot{x}=\frac{\partial x}{\partial t}.$