# Who first solved the classical harmonic oscillator?

There is a question Who solved the quantum harmonic oscillator?, but not one for the classical oscillator. Wikipedia's article Harmonic Oscillator does not have historical information either. So who first solved the classical harmonic oscillator equation?

It was "solved" by Huygens in Horologium Oscillatorum (1673). The scare quotes are there because he never wrote down the equation, and even Newton's laws were not yet explicitly formulated. Huygens considered the motion of pendula, and for simple cases knew the "law of the conservation of living force" (mechanical energy), as Bernoullis later called it, see Mach, History and Root of the Principle of the Conservation of Energy, p. 30. In modern terms, this would be the first integral, or quadrature, of the corresponding equation of motion. With that, he was able to derive the period formula for pendulum motion with small amplitudes, $$T=2\pi\sqrt{\frac{l}{g}}$$ in modern notation, which he also did not use. Here is from Acoustic origins of harmonic analysis by Darrigol, p.351:
"In his effort to treat the vibrating string, John Bernoulli, in a letter of 1727 to his son Daniel and in a paper, considered the weightless elastic string loaded with $$n$$ equal and equally spaced masses... John recognized that the force on each mass is $$-K$$ times its displacement, and solved $$\frac{d^2x}{dt^2} = - Kx$$, thus integrating the equation of simple harmonic motion by analytic methods.
[...] In 1728 Euler began to consider second order equations. His interest in these was aroused partly by his work in mechanics. He had worked, for example, on pendulum motion in resisting media, which leads to a second order differential equation... In a paper of 1739 Euler took up the differential equations of the harmonic oscillator $$\ddot{x} + Kx = 0$$ and the forced oscillation of the harmonic oscillator $$M\ddot{x} + Kx = F\sin(\omega t)$$. He obtained the solutions by quadratures and discovered (really rediscovered, since others had found it earlier) the phenomenon of resonance".