Solve $\epsilon x^3-x+1=0$

I’m trying to find the expansion for the roots of this equation. I’ve found one root as $x\sim 1+\epsilon $. Now considering the dominant balance I want to rescale so that
$\epsilon x^3\sim O(x) \Rightarrow x=O(1/\sqrt\epsilon )$
Setting $x=y(1/\sqrt\epsilon )$ where $y=O(1)$ I get the new equation
Now I want to substitute in $y\sim y_0+\epsilon y_1+\epsilon ^2 y_2+…$ and equate orders of $\epsilon$, but I’m not sure how to deal with the $\sqrt\epsilon$

Help is much appreicated

Solutions Collecting From Web of "Solve $\epsilon x^3-x+1=0$"

Why not to write instead
$$y=a(0)+a(1) \sqrt{\epsilon }+a(2) \epsilon+a(3) \epsilon ^{3/2}+a(4) \epsilon ^2+a(5) \epsilon ^{5/2}+a(6) \epsilon ^3+…$$ Replace $y$ in the equation and start equating the coefficient of each power of $\epsilon$ to $0$. Each power will lead to a linear equation of one coefficient at the time.

If I did not make any mistake, you should arrive at $$y=1-\frac{\sqrt{\epsilon
}}{2}-\frac{3 \epsilon }{8}-\frac{\epsilon ^{3/2}}{2}-\frac{105 \epsilon ^2}{128}-\frac{3 \epsilon ^{5/2}}{2}+\frac{1093 \epsilon
^3}{1024}+ …$$