Remove["Global`*"]
(* Exact solution *)
x[τ] = x0[τ]+ ϵ x1[τ] + (ϵ^2) x2[τ];
x'[τ] = D[x[τ],τ];
x''[τ] = D[x'[τ],τ];
ω = 1 + ϵ ω1 + (ϵ^2) ω2;

Equn = (ω^2) x''[τ] + ϵ ω (((x[τ])^2)-1) x' [τ]+x[τ];
Series[Equn,{ϵ,0,2}]

Plot[{f,ytwo,ythree,yfour},{x,0,10},PlotStyle-> {Black,Red,Blue,Green}]