When to plug in $y = y_0 + \epsilon y_1 + \epsilon^2 y_2 +\cdots$ into the ODE to get the asymptotic series Given a second degree ODE with a small parameter $\epsilon$, 
when can we just plug in $y = y_0 + \epsilon y_1 + \epsilon^2 y_2 + \cdots$ and then solve term by term to get the asymptotic series? 
The singular perturbation case, $\epsilon y'' + a(x) y' + b(x) y = 0$, we can not plug in this, we have to use the WKB theory and plug in $y = e^{y_0/\delta + y_1 + \delta y_2 + \cdots}$. 
And for the duffing equation $y'' + y + \epsilon y^3 = 0$, we have to use multi-scale analysis since the standard perturbation will give us a secular term. 
However, for ODEs like $y'' + 2\epsilon y'+ y = 0$ and $y'' = \epsilon Q(x) y$,we can just plug in the standard series to find the asymptotic. 
For the local analysis without $\epsilon$ in the ODE, there are ways to classify regular points, regular singular points, and irregular singular points, and for each case, we know exactly what type of expansions we should use. So is there a rule to follow when doing the global analysis with $\epsilon$ in the ODE? I guess it will depends on the ODE and also the boundary conditions? 
 A: Side note: assumption of regular perturbation expansion in integer powers of the small parameter is a bit awkwardly restrictive. Assuming merely nonnegative powers of the small parameter allows considerably more flexibility and is still within the realm of regular perturbation theory. 
Anyway, the two problematic cases you pointed out are problematic for entirely different reasons. The first case is because the perturbation itself is singular, which means that the $\epsilon=0$ problem has no solution, so the $y_0$ you would want to construct is already not defined. This generally occurs when setting the small parameter to zero eliminates the highest order term in the differential equation (because now the general solution should be expected to be "lower dimensional", either literally in the linear case or in a loose sense in the nonlinear case). 
The second case is because you want to push the asymptotics to long time scales, but the local approximations you are making have unbounded error on unbounded time scales. On short times there is no problem with regular perturbation expansion, but on long times you need to come to grips with subtle dynamical issues such as periodicity and metastability, which are too complicated for regular perturbation theory to resolve.
