Because of Newton's Law
$$F=ma=m\frac{dv}{dt}=m\frac{d^2x}{dt^2},$$ many physics problems involving force result in second-order equations, where we get the acceleration $a$, rather than the velocity $v$, as a function of position $x$ and time $t$ (and sometimes of velocity).

In this video, we go over the equations describing a mass on a spring, and see how the solutions involve sinusoidal functions.