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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:

$$\epsilon -2y\epsilon \delta_1 + y^2 \delta_1 \epsilon +y \delta = 0 $$

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.

P.S

Cheers!

https://mathhelpboards.com/members/carla1985/

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:

$$\epsilon -2y\epsilon \delta_1 + y^2 \delta_1 \epsilon +y \delta = 0 $$

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.

P.S

**@**Carla1985 you might know how to approach this question if I remember correctly you posted questions on perturbation method and asymptotics.Cheers!

https://mathhelpboards.com/members/carla1985/

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