(Correction for the mass of the spring) Recall that our model of the mechanical oscillator...

(Correction for the mass of the spring) Recall that our model of the mechanical oscillator neglects the effect of the mass of the spring on the grounds that it is sufficiently small compared to that of the mass m. In this exercise we seek to improve our model so as to account, if only approximately, for the mass of the spring. In doing so, we consider the undamped case, for which the equation of motion is mx″ + kx = 0.

Multiplying that equation by dx and integrating, derive the “first integral”

which states that the total energy, the kinetic energy of the mass plus the potential energy of the spring, is a constant.