Answered: "Understanding how to apply dominant balance methodI have..."

High School
Calculus and Analysis
Calculus 2
Differential equations

gabolzm6d

Answered question

2022-04-08

Understanding how to apply dominant balance method
I have the differential equation

ϵ y^{″} - x^{2} y^{'} - y = 0

Answer & Explanation

Alan Bean

Beginner2022-04-09Added 10 answers

Step 1
One method is to assume a power relation

δ = ϵ^{α}

and assume all other coefficients are order 1. This leads to five terms to play with

ϵ^{1 - 2 α}, ϵ^{- α}, 1, ϵ^{α}, 1 .

As you mentioned, you have more freedom and what you choose is up to you. Follow through on each to see how it behaves.
Case 1: Balance the first two - This leads to

α = 1

and the first two terms dominate (some exponential decay).
Case 2: Balance first and third - This leads to

α = \frac{1}{2}

and the second term dominates (

Y \sim c o n s t .

in this scaling).
Case 3: Balance first and fourth - This leads to

α = \frac{1}{3}

and similar behavior as case 2.

Do you have a similar question?

Recalculate according to your conditions!

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