Find the leading order uniform approximation

| August 30, 2017

Question
3. Find the leading order uniform approximation to: εy”+ (e^x)y’ = 1, y(0) = 1, y(infinity) = 0

4. Find the leading order uniform approximation to: εy”+ (1/1+x)y’ + εy = 0, 0<x<1, 0<ε<<1, y(0) = 0, y(1) = 1

Applied Mathematics I – Fall 2015
Homework 8
Due October, 28. 2015.
Note:
• Except for problems that are stated explicitly, all problems are from David Logan
4th Edition.
• Please show all of your work (writing a list of answers is not sufficient).
• Please indicate the people you worked with.
• Please staple your page together.
• 2 random problems will be graded (5 points each).

1. Consider a boundary value problem (this is the same equation as the #5 on HW 7)
εy + y = 0, 0 < ε

1,

y(0) = 0, y(1) = 1.
(a) Show that the regular perturbation method fails.
(b) Use the singular perturbation method to find the leading order uniform asymptotic approximations.
2. Use the dominant balancing method to determine the leading-order behavior of the
roots of
εx3 + x − 2 = 0, 0 < ε
1.
3. Find the leading order uniform approximation to
εy + ex y = 1, y(0) = 1, y(∞) = 0.
(Hint : Taylor series of ex )
4. Find the leading order uniform approximation to
εy + (

1
)y + εy = 0, 0 < x < 1, 0 < ε
1+x
y(0) = 0, y(1) = 1.

(Hint :

1
1+x

≈ 1 − x)

1

1

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