I need to solve the IVP $\frac{d^2u}{dt^2}+[\omega^2+2\epsilon \cos(2t)]u=0$; $u(0)=1$, $\frac{du}{dt}(0)=0$ using the Poincaré method of perturbation. However, I have no idea how to start. We weren't given much theory on this method, and were only given a few examples. However, I have no idea how to extrapolate those examples to this equation since I was given is very misleading to me.
The information given is that $u=u_0+\epsilon u_1(t)+\epsilon^2 u_2(t)+...$, and $\omega^2=n^2 +\epsilon \omega_1 + \epsilon^2 \omega_2+...$. So, if this is the information for the solution, does this mean that time does not need to be perturbed? And what does this expansion for $\omega$ mean and why is it justified? Also, should it just be substituted into the equation and then the equation simply needs to be solved without perturbing time? Very weird...
Some help would really be appreciated!