Are differential equations beyond coordinates? In Physics when we write down Newton's second law, the differential equation we have is coordinate agnostic. Meaning, we can put in any coordinates into the equation and get a second order DE which models the motion of the object. Now, an interesting point here is with the coordinate agnosticism that is going on: we are saying that the differential equation exists beyond a statement in any particular coordinate system as a motion of some body in space.
However, when we learn what differential equations are in elementary courses, we see them as equations in fixed coordinates of form :
$$ F \left( x,y,y',y'',\dots,y^n \right) = 0$$
Of course, even here it is reasonable to talk about substitutions by setting $x$ as a function of another variable $\chi$. The point I want to emphasize is , there is no aspect of a fundamental coordinate independent idea (such as motion said previously) going on here.
So, how do these two viewpoints relate? I am looking for an overview answer.
 A: tl; dr: The coordinate-invariant expression of an ordinary differential equation is a vector field on a manifold.

$\newcommand{\Reals}{\mathbf{R}}$In many circumstances, such as classical physics, the configuration space or set of possible states of a system is a manifold $M$: The position of a particle in the plane $\Reals^{2}$ is modeled by $\Reals^{2}$ itself; the angular position of a pendulum is modeled as an angle, i.e., a point on a circle; the position and spatial orientation of a gyroscope are modeled by $\Reals^{3} \times SO(3)$; the positions of $N$ identical but distinct particles in three-space are modeled by points of $(\Reals^{3})^{N} \setminus \Delta$, with $\Delta$ denoting the set of $N$-tuples (of points in $\Reals^{3}$) with at least two points equal; etc.
A "physically-reasonable" first-order autonomous equation $F(y, y') = 0$ can in principle be written in the form $y' = X(y)$. That is, if we interpret $y$ as "the state of the system at time $t$" then the state $y$ of the system determines the infinitesimal motion $y'$. The state $y$ may be viewed as a point of $M$, the "dynamical law" $X$ may be interpreted as a vector field on $M$, and a solution of the flow equation $y' = X(y)$ is geometrically an integral curve of $X$, namely the image of a maximal smooth path $y$ defined in some open interval $I$ of real numbers containing $0$ and satisfying
$$
y'(t) = X(y(t))\qquad \text{for all $t$ in $I$.}
$$
A first-order time-dependent ODE $y' = X(t, y)$ may be viewed as an autonomous vector field on $\Reals \times M$. That is, we prepend a real time coordinate $t$ to our state space, define a vector field $G$ on $\Reals \times M$ by $G(t, y) = (1, X(t, y))$, and look for solutions of the equation $Y' = G(Y)$.

Similarly, a second order autonomous equation can be written, in principle, in the form $y'' = X(y, y')$. Physically, the position and velocity determine the acceleration. This second-order equation in $y$ can be written as a first-order system for the ordered pair $(y, y')$ as $(y, y')' = (y', y'') = (y', X(y, y'))$. Invariantly, the pair $(y, y')$ comprises a position $y$ and a velocity $y'$, and may be viewed as a single element of the tangent bundle $TM$.
That is, an autonomous second-order equation on $M$ may be viewed as an autonomous first-order equation on $TM$, the state space for position-and-velocity. (I believe physicists call this the Lagrangian formulation of mechanics.)
Time-dependent second-order equations, and higher-order equations, may be described recursively using the same ideas.
