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  • $\begingroup$ What is the parameter range of k ? $\endgroup$
    – Ulrich Neumann
    Commented Apr 1 at 11:59
  • $\begingroup$ Would the quasi-static/subhorizon approximation work in your case? Getting the full analytical solution might be very difficult, that's why there are the Boltzmann codes out there ;-) $\endgroup$
    – Hans Olo
    Commented Apr 1 at 13:32
  • $\begingroup$ Why do you expect there to be an analytic solution to this equation? Analytic solutions are rare, especially for nonlinear equations like yours. I'd be surprised if there was one here, unless you can find some conserved quantities or something to simplify the differential equation. Of course, that's a math problem rather than a Mathematica one. $\endgroup$
    – march
    Commented Apr 1 at 15:15
  • $\begingroup$ @UlrichNeumann. $-5<k<5$. $\endgroup$
    – Dr. phy
    Commented Apr 1 at 15:18
  • $\begingroup$ @Dr.phy Your ode seems to be singulaer near t==0: {Coefficient[eq, x''[t]] /. t -> 0, Asymptotic[eq /. x''[t] -> 0, t -> 0]} (*{-3, -((4 x[0])/(E^2 t)) + (3 Derivative[1][x][0])/t}*) $\endgroup$
    – Ulrich Neumann
    Commented Apr 1 at 15:59