Asymptotic mathcing for a first order differential equation

In summary, the conversation discusses a first-order differential equation with a boundary value and the use of asymptotic matching to study its behavior as e tends to 0. The conversation explains that when x is not too large, the term ex^2 is negligible and an approximate equation for y can be found. However, when x is large, ex^2 is not negligible but 1/x^2 is, leading to a different approximate equation. The conversation also addresses the confusion around the upper edge of the left region and clarifies that the region of validity for the left solution is actually smaller than initially thought. The conversation concludes with a clarification on the use of "e" in the context of this problem.
  • #1
hanson
319
0
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:
Mathematics news on Phys.org
  • #2
I'm confused - [tex]e[/tex] is usually a constant
[tex]e~=2.71828182845904523536028747135266249775724709369995...[/tex]
 
  • #3
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...
 
  • #4
e= epsilon: [itex]\epsilon[/itex]
 
  • #5
yes, HallsofIvy! Thank you.

Do you mind helping me for this question?
 

Related to Asymptotic mathcing for a first order differential equation

What is asymptotic matching for a first order differential equation?

Asymptotic matching is a method used to solve first order differential equations with small parameters. It involves finding an approximate solution by matching the terms of the differential equation as the parameter approaches zero.

Why is asymptotic matching useful?

Asymptotic matching allows us to find an approximate solution to a first order differential equation without having to solve it exactly. This can be helpful when the equation is difficult or impossible to solve exactly.

What is the process of asymptotic matching?

The process of asymptotic matching involves expanding the solution to a first order differential equation in powers of the small parameter, and then matching the terms of the equation to find the dominant behavior of the solution as the parameter approaches zero.

What are the limitations of asymptotic matching?

Asymptotic matching is only applicable to first order differential equations and may not provide an accurate solution for all values of the parameter. It also requires some knowledge of the behavior of the equation, as well as the ability to manipulate and match terms.

Can asymptotic matching be used for higher order differential equations?

No, asymptotic matching is only applicable to first order differential equations. For higher order differential equations, other methods such as perturbation theory or numerical methods may be used to find approximate solutions.

Similar threads

I Partial Differential Equation solved using Products
    • General Math
Replies
2
Views
459
I A doubt about the multiplicity of polynomials in two variables
    • General Math
Replies
8
Views
1K
I Is there a name for this sort of differential equation?
    • General Math
Replies
2
Views
1K
MHB Solve Differential Eq: xe^-1/(k+e^-1) for x, k, t
    • General Math
Replies
2
Views
1K
Solve the given first order differential equation
    • Calculus and Beyond Homework Help
Replies
8
Views
821
I Solving Differential Equation Using Reduction of Order
    • Differential Equations
Replies
2
Views
1K
Solve these two coupled first-order differential equations and sketch the flow
    • Advanced Physics Homework Help
Replies
2
Views
1K
I Fundamental matrix of a second order 2x2 system of ODEs
    • Differential Equations
Replies
2
Views
2K
Reduction of order for Second Order Differential Equation
    • Calculus and Beyond Homework Help
Replies
2
Views
355
MHB [ASK] Equation of a Circle in the First Quadrant
    • General Math
Replies
4
Views
1K

Hot Threads

Recent Insights

Back
Top