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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?

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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:

"The first intimation that harmonic (sine-like) motion plays a basic role in acoustics is found in Christiaan Huygens's theory of musical strings. In his celebrated Horologium Oscillatorium of 1673, Huygens showed that the pendulous motion of a body sliding down on a cycloid was harmonic and isochronous. Around that time, he also understood that the force responsible for this motion was proportional to the distance traveled by the body from the point of equilibrium. Probably noticing that a similar circumstance held in the case of a tense, weightless elastic string loaded with one mass in the middle, he derived the oscillation frequency as a function of tension, length, and mass. The reasoning implied harmonic oscillations for the loaded string. He also sketched a generalization to a string loaded with several masses, in which he assumed all the masses to perform harmonic oscillations of the same frequency and phase."

Taylor, who was studying vibrating strings in 1713, had the benefit of Newton's mechanics. Still, he did not write down the equation, but used pendulum analogies, and realized that the sine shape was a solution for the string. Johann Bernoulli followed Taylor's lead, and represented strings as connected masses. In a 1727 letter to his son Daniel he explicitly wrote the harmonic oscillator equation for each and integrated it analytically.

In print, the first modern treatment of the harmonic oscillator is Euler's De Novo Genere Oscillationum (presented 1738-9, published 1750). He solved in quadratures not only the equation of the free oscillator, but also of the oscillator driven by harmonic force. This was the first analytic treatment of resonance, see Kline, Mathematical Thought From Ancient to Modern Times, v.2, pp. 479-482:

"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".

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