Solving (x^2)y`` -xy`+y=0: Tips for Beginners

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Homework Help Overview

The discussion revolves around finding a basis of solutions for the differential equation (x^2)y`` -xy`+y=0, which is identified as an "Euler-type" equation. Participants are exploring methods to approach this problem, particularly for beginners.

Discussion Character

  • Exploratory, Conceptual clarification

Approaches and Questions Raised

  • The original poster expresses uncertainty about where to start and seeks initial guidance. Some participants inquire about the use of power series methods, while others suggest a specific substitution method involving y= xr for an unknown r.

Discussion Status

The conversation is ongoing, with some participants providing hints and suggestions to guide the original poster. There is an acknowledgment that the proposed substitution could be a starting point, but no consensus has been reached on the overall approach.

Contextual Notes

The original poster has indicated a lack of familiarity with power series methods, which may influence their understanding of the problem. The discussion also highlights that the equation's structure is characteristic of Euler-type equations.

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for the following question:
find a basis of solutions for
(x^2)y`` -xy`+y=0


my problem:
i haven't a clue on where to start~
can someone give some tips for starters?
 
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Have you learned power series methods (Frobenius) for solving linear ODEs?
 
no, not yet
 
It's not necessary to use power expansions here. This is an "Euler-type" or "equipotential" equation- the degree of x in each term is equal to the order of the derivative.

Hint- let y= xr for unknown r, then find r so that the equation is satisfied.

(That will get you started, but that's actually the easy part!)
 
ok, i'll try that!
 

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