I have a differential equation of the form $$ y'' + p(x) y' + (1-\epsilon^2 q(x)) y = 0\ , $$ where $y$ is also a function of $x$.
I'm supposed to apply a perturbation method via a regular expansion $$ y = y_0 + \epsilon y_1 + \epsilon^2 y_2 + \dots $$ such that I don't need to know $p(x)$ in order to calculate the dominant term $y_0$. I don't see how is that possible, since I would end up with an equation like $$ y_0'' + p(x) y_0' + y_0 = 0 $$
I have never seen an equation where the small parameter appears as $\epsilon^2$ instead of $\epsilon$, so I don't know if that is the key for solving this the correct way. Any help on this would be appreciated.