I'm trying to find first two non-zero terms in the asymptotic expansion for each of the three roots of
$\epsilon x^3 - x^2 + x -\epsilon ^{1/2} = 0$
as $\epsilon \rightarrow 0^{+}$.
The naive expansion of the form $x=x_0 +\epsilon x_1 + O(\epsilon^2)$ gave only two roots. How do I attack this? I read about possibly using a Newton-Kruskal diagram but I don't really understand the argument made in this question Asymptotic expansions for the roots of $\epsilon^2x^4-\epsilon x^3-2x^2+2=0$ .
I want to rescale the equation by setting $X = x\epsilon^{\nu}$ but, I'm not sure on how to balance this? Do I need to eliminate the $\epsilon^{1/2}$ term?
Thanks for any help in advance.
If you don't want to deal with the Newton polygon or Laurent/Puiseux series, set $y=\epsilon x$ and $\epsilon=\zeta^2$. Then it is just as well to find series solutions to $$y^3-y^2+\zeta^2 y-\zeta^5=0\text{.}$$ The naïve method will work for this equation.