If you don't mind can you check this too , I wrote that it's uniformly valid as the second term is much less than the first term and it becomes non uniform when x^2 << O( epsilon^-1)
If you don't mind can you check this too , I wrote that it's uniformly valid as the second term is much less than the first term and it becomes non uniform when x^2 << O( epsilon^-1)
You're not answering this part correctly, though you have the somewhat basic understanding of what you need to mention here.
When x2 is of the same magnitude of as ϵ−1, that's when the term ϵ2x2 starts behaving like the leading order order term. This is bad, since it means there is a substantial contribution coming from here to the leading order, which pollutes the solution. This term is known in the biz as a secular term because it grows with the independent variable.
Hence, the expansion is nonuniform for all x, and becomes bad when x2=O(ϵ−1). Otherwise, the expansion is good enough when x2≪O(ϵ−1).
This is why in practice when we are interested in long-time asymototics, this regular perturbation is somewhat rubbish since it does not hold when x is sufficiently large ('sufficiently large' here differs from context to context). One way to ensure our solution is valid for as much of the domain as possible is to decrease ϵ further and further. But this can be upgraded a whole lot using the method of multiple-scales asymptotic analysis (which I hope you are going to learn if you're already on the regular perturbation expansion!) whereby we do not need to decrease epsilon as much to obtain an accurate solution over a sufficiently large domain.