Hello!
For a coupled two-body oscillator we write the general solution as:
x1(t)=C1-Cos[ω-t+ψ1-]+C1+Cos[ω+t+ψ1+]
x2(t)=C2-Cos[ω-t+ψ2-]+C2+Cos[ω+t+ψ2+]
Where we determine C1-/C2- and C1+/C2+ from the normal mode condition.
We call ψ1-=ψ2-=ψ- and ψ1+= ψ2+=ψ+, and we end up with 4...