How do I know what scales to look for for asymptotic expansion? E.g. cubic case?
How do I check for their validity?
Such as in the example here:
http://www.macs.hw.ac.uk/~simonm/ae.pdf
where they find that
$x^3-x^2-(1+\epsilon)x+1=0$
has double root at $x=1$ and it suggests them to try an expansion in powers of $\epsilon^{1/2}$.
But one could want to solve e.g.
$x^3-5x^2+4x+\epsilon=0$
Given here:
http://sycon.rutgers.edu/~speer/528s17/Perturbation-DG.pdf
which has roots $x=0,x=1,x=4$.
In the first equation you have $$ (x+1)(x-1)^2=ϵx\iff (x-1)^2=\frac{ϵx}{1+x} $$ Close to $x=1$ this gives approximations for the roots there of $$ x=1\pm\sqrt{\frac{ϵx}{1+x}}\approx 1\pm\sqrt{\fracϵ2} $$ For the root close to $-1$ you get similarly $$ x=-1+\frac{ϵx}{(1-x)^2}\approx -1+\fracϵ4 $$
For the second equation goes the same, $$ x(x-1)(x-4)=-ϵ $$ gives approximations close to the roots of the undisturbed equation of $$ -\fracϵ4,~~1+\fracϵ3,~~ 4-\fracϵ{12} $$