I started a lecture on differential equation with following example. If a body is moving in a straight line in plane with constant speed, how can we describe this motion mathematically?
To answer this, put a coordinate system in plane. Then the path the body is following is given by equation of straight line $ax+by+c=0$. If the coordinate system is changed then equation of line also gets changed; it will be as $a'x+b'y+c'=0$. One can observe that for both the coordinate systems, the path of motion is solution of the differential equation $\frac{d^2y}{dx^2}=0$. Then I said
The motion of body is expressed by the differential equation $\frac{d^2y}{dx^2}=0$, rather than linear equation, since it is coordinate-free.
The following question raised then in the class:
What $y$ and $x$ represents in the differential equation? Are they coordinates?
I couldn't answer this. Can one help me? What is the correct way to say the equation of motion of body (in straight line, with constant speed), which is coordinate-free?
Given that the $Oxy$ coordinate system is an inertial reference frame, the equation of motion for the body (whose mass is $m$) is given by Newton's $2$nd law of motion, i.e. $$\sum F=m\frac{d^2y}{dx^2}=0$$ and its solution reads: $\frac{d^2y}{dx^2}=0\Leftrightarrow \frac{dy}{dx}=a\Leftrightarrow y=ax+b$ which represents a straight line in $Oxy$ (that is the trajectory of the free particle in an inertial reference system), with $a$, $b$ being the integration constants to be determined from the two initial conditions: the position $x(t_0)$ at $t=t_0$ and the velocity $u(t_0)=\frac{dx}{dt}\Big|_{t_0}$ at $t=t_0$.
On the other hand, if the coordinate system is changed from $Oxy$ to $O'x'y'$, and the new coordinate system $O'x'y'$ keeps a constant velocity with respect to $Oxy$, i.e. if the new reference frame is again an inertial reference frame, then in the $O'x'y'$ coordinate system, the equation of motion is given again by Newton's $2$nd law of motion, but now with respect to the new coordinates $x',y'$, i.e.: $$\sum F'=m\frac{d^2y'}{dx'^2}=0$$ and its solution reads: $\frac{d^2y'}{dx'^2}=0\Leftrightarrow \frac{dy'}{dx'}=a\Leftrightarrow y'=a'x'+b'$ which represents a straight line in $O'x'y'$ (i.e. the trajectory of the free particle in an inertial reference system), with $a'$, $b'$ being the integration constants (now, with respect to the coordinates $x'$, $y'$).
P.S.: Newton's $2$nd law of motion actually provides a coordinate free description of the motion of a body, as long as we keep ourselves confined to inertial observers: $$ \sum\vec{F}=m\vec{a} $$ where, $\vec{F}$ is the resultant force exerted on the body and $\vec{a}$ is the acceleration of the body, both measured with respect to an inertial reference frame.