The equation of motion (E of M) of a system is an equation which expresses the fundamental laws of motion in terms of only (a) the degrees of freedom, (b) derivatives of the degrees of freedom, and (c) constants characterizing the system.
In the example of the SHO, the fundamental law of motion is Newton's law. Because our particle is free to move along the x-axis only, we need consider just the x-components of the motion. Newton's law states for this motion,
The only force acting along the
x-direction (because the rod is frictionless) comes from the spring and
is in direct proportion to the stretch in the spring
through the spring constant k. Moreover, when the
spring is stretched, 
(and 
), the force is
backwards along the x-axis. Therefore, 
. Thus, we have
The above equation does not yet qualify as an equation of motion
because the acceleration 
does not fall into the allowed
categories (a-c). We amend this simply by replacing the acceleration
in the x-direction by its mathematical equivalent 
and, thereby, derive the equation of motion for the SHO,
Eq. (5) now qualifies as an equation of motion because
all terms fall into the above categories: x(t) is in (a), 
is in (b), and -k, 
and m are in (c).
