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.
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.