Asymptotic mathcing for a first order differential equation

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Discussion Overview

The discussion revolves around the asymptotic matching for a first-order differential equation, specifically analyzing the behavior of the solution as a small parameter, referred to as e, approaches zero. The context includes exploring the validity of approximations for different ranges of x.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant presents the differential equation and proposes asymptotic matching to analyze the solution's behavior as e tends to zero.
  • The participant suggests that for small x, the term ex^2 is negligible, leading to an approximate equation for y.
  • For large x, the participant argues that ex^2 is significant while 1/x^2 is negligible, resulting in a different approximate equation for y.
  • The participant expresses uncertainty about the boundary between the left and right regions of validity, questioning why the left region is valid for x << e^(-1/2) as e approaches zero, but actually extends to e^(-1/3).
  • Another participant expresses confusion regarding the notation of e, typically understood as a constant.
  • A clarification is provided that e refers to a small perturbation, with a suggestion to use the Greek letter epsilon instead.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the boundary conditions for the regions of validity of the approximations, and there is confusion regarding the notation used for the small parameter.

Contextual Notes

There is a lack of clarity regarding the assumptions made about the parameter e and its implications for the asymptotic analysis. The discussion also highlights the need for precise definitions in mathematical notation.

Who May Find This Useful

Readers interested in differential equations, asymptotic analysis, and perturbation methods may find this discussion relevant.

hanson
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The first-order differential equation
[tex]y' +(ex^2+1+1/x^2)y=0[/tex], with boudary value y(1) =1

Using, asymptotic mathcing to study the behaviour of the sltion as e tends to +0, when x is not too large, the term [tex]ex^2[/tex] is negligible so an approximate equation for y is
[tex]y'_L +(1+1/x^2)y_L=0[/tex].

When x is large, [ tex ]ex^2 [ /tex ] is not negligible but [tex]1/x^2[/tex] is. Therefore, an approximate equation valid as x tends to infinity is [tex]y'_R +(ex^2+1)y_R=0[/tex].

I think that te upper edge of the left region would be the largest value of x for which ex^2 is still small compared with 1. This would suggest that the left region consists of those x for which x<<e^-(1/2) as e tends to +0. But actually the region of validity of the left solution is e^(-1/3) (e tends to +0)...Can anyone explain this to me??

Sams as the right region...Please kindly help
 
Last edited:
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I'm confused - [tex]e[/tex] is usually a constant
[tex]e~=2.71828182845904523536028747135266249775724709369995...[/tex]
 
sorry, I am referring e to be a small pertubation introduced...
It would better to use a Greek word for it...
but I have forgotten the word...
 
e= epsilon: [itex]\epsilon[/itex]
 
yes, HallsofIvy! Thank you.

Do you mind helping me for this question?
 

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