Question: Find the rescalings for the roots of $$\epsilon^5 x^3 - (3 - 2\epsilon^2 + 10\epsilon^5 - \epsilon^6)x^2 + (30 - 3\epsilon -20 \epsilon^2 + 2\epsilon^3 + 24\epsilon^5 - 2\epsilon^6 - 2\epsilon^7)x - 72 + 6\epsilon + 54\epsilon^2 = 0$$ and hence find two (non-trivial) terms in the approximation for each root using expansion method.
My approach: Substituting $\epsilon = 0$ in the original equation and solving the resultant quadratic equation gives us $x = 4$ and $x = 6$ as the unperturbed roots. This implies that the regular roots of the original equation would be of the form: $$x_1 = 4 + x_{11}\epsilon + x_{12}\epsilon^2 + x_{13}\epsilon^3 + \cdots, \text{ and}$$ $$x_2 = 6 + x_{21}\epsilon + x_{22}\epsilon^2 + + x_{23}\epsilon^3 + \cdots.$$
Now, to find the singular root of the original equation, denoted by $x_3$, we need to establish a suitable scaling $x = \delta(\epsilon)X$, where $X$ is strictly of order 1 as $\epsilon \to\infty$. Doing dominant balances across different ranges of $\delta$, we find that $\delta = 1$ and $\delta = \epsilon^{-5}$ are the only feasible values. $\delta = 1$ would yield the two regular roots $x_1$ and $x_2$, and $\delta = \epsilon^{-5}$ would yield the singular root $x_3$.
We substitute $x = \delta X = \epsilon^{-5}X$ in the original equation to yield the following rescaled equation: $$X^3 - (3-2\epsilon^2 + 10\epsilon^5 - \epsilon^6)X^2 + (30\epsilon^5 - 3\epsilon^6 - 20\epsilon^7 + 2\epsilon^8 + 24\epsilon^{10} - 2\epsilon^{11} - 2\epsilon^{12})X - 72\epsilon^{10} + 6\epsilon^{11} + 54\epsilon^{12} = 0.$$
Substituting $\epsilon = 0$ in the above-determined rescaled equation, it simplifies to the following equation: $$X^3 - 3X^2 = 0,$$ which implies that either $X = 0$ or $X = 3$. $X$ cannot be equal to $0$, since we assumed that $X = \mathrm{ord}(1)$ when $\epsilon \to 0$. Thus, $X = 3$. This implies that the rescaled singular root is of the form: $$X_3 = 3 + x_{31}\epsilon + x_{32} \epsilon^2 + x_{33}\epsilon^3 + \cdots.$$ Substituting this expression into the rescaled equation and equating the coefficients of $\epsilon, \epsilon^2$ and $\epsilon^3$, we find that $x_{31} = 0$, $x_{32} = -2$ and $x_{33} = 0$, which implies that the rescaled singular root is $$X_3 = 3 - 2\epsilon^2 + \mathcal{O}(\epsilon^4).$$ This in turn implies that the singular root is $$x_3 = \frac{3}{\epsilon^5} - \frac{2}{\epsilon^3} + \mathcal{O}(\epsilon^{-1}).$$
In a similar manner, the exact expressions of the two regular roots can be determined: $$x_1 = 4 + \epsilon -\frac{2}{3}\epsilon^3 + \mathcal{O}(\epsilon^4),$$ $$x_2 = 6 - 2\epsilon +\frac{2}{3}\epsilon^3 + \mathcal{O}(\epsilon^4).$$
Can someone please check if this solution is correct? Also, if possible, does anyone know how to solve this problem using iteration in an efficient manner?