how do i find the bifurcation for this:

Question

d^x/dt^2 = -.3 x - (x - 1)/((x - 1)^2 + a^2)^(3/2) - (x + 1)/((x + 1)^2 + a^2)^(3/2)

we've tried using mathematica but it just runs forever...

Answer 1

Well by inspection,

dx/dt=-.3/2 x^2 - 1/2((x-1)^2+ a^2)^.5+1/2 (((x+1)^2+a^2)^.5

and then

x= -1/20x^3- 1/3 ((x-1)^2+ a^2)^3/2 + you finish it.

Did I miss something?