Understanding Well-Posed Differential Equations: Tips and Techniques

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Discussion Overview

The discussion revolves around the criteria for determining if a differential equation is well-posed, focusing on the existence and uniqueness of solutions. Participants explore theoretical foundations and resources related to ordinary differential equations (ODEs).

Discussion Character

  • Exploratory, Technical explanation, Conceptual clarification, Homework-related

Main Points Raised

  • One participant questions how to determine if a differential equation is well-posed, noting that it seems to depend on author and convention.
  • Another participant suggests that to establish existence and uniqueness, one should refer to general theorems such as the Picard-Lindelof theorem and the Peano existence theorem.
  • A participant expresses uncertainty about the clarity of the original question, indicating a sense of ambiguity.
  • There is a recommendation to consult Ross' Differential Equations book, with a distinction made between his larger theoretical book and a smaller practical one.

Areas of Agreement / Disagreement

Participants express uncertainty about the clarity of the question regarding well-posedness and acknowledge that definitions may vary by author. There is no consensus on a singular approach to determining well-posedness.

Contextual Notes

Participants note the dependence on specific theorems for establishing existence and uniqueness, indicating that the discussion may hinge on the definitions and conventions used in different contexts.

Who May Find This Useful

Readers interested in differential equations, particularly those seeking to understand the concepts of well-posedness and relevant theorems.

member 428835
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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