For those familiar with the WKB method

[Poirot1]

let x be the parameter in some interval and let lambda tend to infinity. Should I treat x as of the same order as constants, i.e x=O(1)?

 

[Sudharaka]

let x be the parameter in some interval and let lambda tend to infinity. Should I treat x as of the same order as constants, i.e x=O(1)?

Hi Poirot, :)

I am sorry but I don't understand your question. Are you taking about the WKB approximation method? Can you please elaborate further?

Kind Regards,
Sudharaka.

 

[Poirot1]

Hi Poirot, :)

I am sorry but I don't understand your question. Are you taking about the WKB approximation method? Can you please elaborate further?

Kind Regards,
Sudharaka.

This is indeed what I am talking about Sudharaka. I have two questions for you- one which can be best illustrated by an example. For large t, find wkb approximation of

$y''-y(t^{2}x^{2}+tx^{-1})=0$ for x not zero

We do this by substituting y=exp(...) (I'm sure you're familiar with this) and finding dominant balances. In the second dominant balance equation, I had potential driving terms t and t/x, and I thought both should be retained in the dominant balance as they are both of O(t). But the answer was wrong and it was clear that I should only retain t.
Why?

My second (and original) question is if I have to choose between x and a constant between retained in the dominant balance, ought I to retain both? I.e. x=O(1)