I'm trying to find the roots to the following equation:
$t^5 - \epsilon t^3 + \epsilon^3 = 0$
as $\epsilon \rightarrow 0$.
From expansion $t= \epsilon^{\alpha}t_1 + \epsilon^{2\alpha}t_2 + \mathcal{O}(\epsilon^{3\alpha})$ this gives me two of the roots:
$t= \epsilon^{1/2} -\frac{1}{2}\epsilon + \mathcal{O}(\epsilon^{3/2})$
$t= -\epsilon^{1/2} -\frac{1}{2}\epsilon + \mathcal{O}(\epsilon^{3/2})$
I can't find the other 3 roots. Can someone please help?