17.1.3—A = –ω2x and recall and use, as a solution to this equation, x = x0 sin ωt

A sinusoidal solution x=x₀ sin(ωt+φ) satisfies a=−ω²x, with amplitude x₀, angular frequency ω and phase φ.

Use the initial phase to match the starting position and direction; differentiate to obtain velocity and acceleration.

If the oscillator starts at equilibrium moving positive, x=x₀ sinωt is a natural phase choice.

x₀ is amplitude, not an arbitrary displacement, and omitting phase can give the wrong initial condition.