Understanding Coupled Oscillator Equations of Motion

[Piano man]

Hi, this is a fairly basic part of the whole coupled oscillators area, but I don't really get it.
My problem is with the equations of motion of a coupled oscillator:
F_A=-kx_A -2k'x_A
and
m\ddot x_A = -kx_A -k(x_A-x_B)

Everywhere I've read seems to take it as intuitive, but I don't see how.
Can someone explain it please?

 

[diazona]

It's been a while since I did this sort of thing, but might you be missing a term or two? Those equations don't look quite right. (Also, just to be clear, what is each of your variables supposed to mean?)