Understanding Well-Posed Differential Equations: Tips and Techniques

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SUMMARY

This discussion focuses on determining whether ordinary differential equations (ODEs) are well-posed, emphasizing the necessity of existence and uniqueness of solutions. Key theorems mentioned include the Picard-Lindelof theorem, which guarantees uniqueness, and the Peano existence theorem, which establishes existence for general ODEs. Participants recommend consulting Ross' Differential Equations book, specifically the comprehensive version that covers theoretical aspects rather than just practical applications.

PREREQUISITES
  • Understanding of ordinary differential equations (ODEs)
  • Familiarity with the Picard-Lindelof theorem
  • Knowledge of the Peano existence theorem
  • Basic concepts of mathematical analysis
NEXT STEPS
  • Study the Picard-Lindelof theorem in detail
  • Explore the Peano existence theorem and its implications
  • Read Ross' Differential Equations book for a comprehensive understanding
  • Investigate additional resources on well-posedness in differential equations
USEFUL FOR

Mathematicians, students of differential equations, and educators seeking to deepen their understanding of well-posed problems in mathematical analysis.

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How to tell if a differential equation is well posed? I understand this means a solution exists and is unique. How would one determine if an ODE is well posed?
 
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joshmccraney said:
How to tell if a differential equation is well posed? I understand this means a solution exists and is unique.

I don't know. Does it? It seems to depend on author and convention.

How would one determine if an ODE is well posed?

To determine existence/uniqueness, you'll need to have a general theorem establishing them. The Picard-Lindelof theorem and the Peano existence theorem are two popular ones which show existence (and uniqueness for Picard-Lindelof) for quite some general ODE's.
 
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Shoot, I had a gut feeling there was some ambiguity in asking the question!

And thanks! I'll look into your suggestions! I just needed a place to start.
 
Do check out Ross' Differential Equations book. It's one of the best books on the topic. (Do get his large book which covers the theory, and not his small book by almost the same name which covers just the practical stuff).
 
Sounds good!
 

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