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

Professor Strang’s Calculus textbook (1st edition, 1991) is freely available here.

Subtitles are provided through the generous assistance of Jimmy Ren.