I have this question from Murdock's textbook called: "Perturbations: Methods and Theory": Use rescaling to solve: $\phi(x,\epsilon) = \epsilon x^2 + x+1 = 0$ and $\varphi (x,\epsilon) = \epsilon x^3+ x^2 - 4=0$.
I'll write my attempt at solving these two equations, first the first polynomial.
According to the book I first need to find an undetermined gauge that solves the above equation when $\epsilon = 0$, which means I need to guess $x \approx -1 +y \delta_1(\epsilon)$, now I plug it back to the above equation to find the following equation: $$\epsilon -2y\epsilon \delta_1 + y^2 \delta_1^2 \epsilon +y \delta_1 = 0 $$
Now my problem is in finding a suitable gauge for $\delta_1(\epsilon)$ that solves the above last equation, it should be proportional to some power of $\epsilon$, i.e $\epsilon^r$, such that higher powers of $\epsilon$ in the last equation get neglected and we can find a solution for $y$ which gives a non-zero solution to $y$, but I don't see it.
Any help?