Homogenous Ordinary Differential Equation

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Homework Help Overview

The discussion revolves around a homogeneous ordinary differential equation of the form x²y'' - (x² + 2x)y' + (x + 2)y = 0. Participants are tasked with determining the Wronskian given known solutions at a specific point.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • One participant attempts to simplify the equation by dividing out the x² term to transform it into a linear system, expressing uncertainty about the next steps to find the characteristic solution. Another participant references Abel's theorem as a means to find the answer. There is also a question raised regarding the applicability of the complementary general solution equations for differential equations with variable coefficients.

Discussion Status

The discussion has seen some participants providing guidance and referencing theorems, while others are exploring different interpretations of the problem. There is no explicit consensus, but some productive directions have been suggested.

Contextual Notes

Participants are navigating the complexities of differential equations with variable coefficients, which may influence their approaches and understanding of the problem.

sndoyle1
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Homework Statement



x2y"-(x2+2x)y'+(x+2)y=0

known solutions:

y1(2)=2
y1'(2)=1
y2(2)=2e2
y2'(2)=3e2

Determine the wronskian

Homework Equations



yc=C1er1x+C2er2x

I also know how to find the wronskian via a determinant

The Attempt at a Solution



I have tried to divide out the first x2 term to make this a linear system, not sure how to simplify after this to find the characteristic solution.
 
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I found the answer using Abel's theorem.
 
nvm: I see you found your answer.

Isn't the complementary general solution equations you provided for differential equations with constant coefficients, rather than coefficients that are functions of a variable?
 
Last edited:
you are correct. I was confused with the question and obviously drew a blank.
 

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