These equations have 2nd derivatives because acceleration is in Newton's Law F = ma
The key model equation is (second derivative) y ' ' = MINUS y or y ' ' = MINUS a^2 y

There are two solutions since the equation is second order. They are SINE and COSINE
y = sin (at) and y = cos (at) Two derivatives bring back sine and cosine with minus a^2

The next step allows damping (first derivative) as in my ' ' + dy ' + ky = 0 How to solve?
Just try y = e^at !! You find that ma^2 + da + k = 0 Two a's give two solutions: good