First Order Perturbation Theory help

First, I apologize for the bad formatting; I'm not sure how to use the system well. In these equations, square brackets indicate subscripts.

I'm having trouble understanding how to use perturbation theory, and was wondering if anyone here could help.

The problem I'm stuck on is to use first order perturbation theory to analyze the equation of a nonlinear, driven oscillator with equation of motion

x"+x+ϵx^3 = gcos(ωt), with ϵ <<1.

I know that I need to find a solution x(t)=x[0](t) + ϵx[1](t) +... but I'm not sure how to proceed.

Finally, I then need to show that x[0](t) and x[1](t) satisfy the linear differential equations of

x''[0]+x[0] = gcos(ωt)

x''[1]+x[1] = -(x[0])^3

The second part might be easy once I can figure out the first part, so that's the first part is more important.

Thank you for your help!