ODE with Parameter: Is \phi(x,0) a Solution to y' = f(x,y,0)?

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Homework Help Overview

The discussion revolves around an ordinary differential equation (ODE) involving a parameter, specifically examining whether a function \(\phi(x,0)\) serves as a solution to the ODE when the parameter is set to zero. The original poster is seeking clarification on the implications of a theorem related to the dependence of ODE solutions on parameters.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to understand the relationship between the solution \(\phi(x,\epsilon)\) and its evaluation at \(\epsilon=0\). Some participants suggest referencing a theorem related to parameter dependence in ODEs, while others express concerns about the relevance of the homework to lecture content.

Discussion Status

The discussion is ongoing, with participants exploring the implications of the theorem mentioned and questioning the alignment of the homework with their course material. Some guidance has been offered regarding the differentiation of \(\phi(x, \epsilon)\) with respect to \(x\) and evaluating at \(\epsilon=0\).

Contextual Notes

Participants note a lack of relevant course materials, such as textbooks, which may hinder their ability to reference the theorem effectively. Additionally, there is an acknowledgment of the professor's tendency to assign homework that does not directly correlate with lecture content.

dipole
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Homework Statement



In a HW assignment, I'm given the ODE

[itex]y' = f(x,y,\epsilon)[/itex]

and that [itex]y = \phi(x,\epsilon)[/itex]is a solution to this equation.

I'm then asked, is [itex]\phi(x,0)[/itex] a solution to the equation

[itex]y' = f(x,y,0)[/itex]

This result is used for the second part of the problem, and in the question I'm told I can just quote a well known theorem to explain why it's true, but I have no idea what theorem that might be. Any ideas, or maybe how to even prove it?
 
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There is a theorem on the dependence of ODE solutions on parameters. I am sure it has been covered in your course of ODEs.
 
You would think so, but the professor constantly assigns HW that has little relevance to what we've actually done in lecture. Also we have no textbook to use as a reference.
 
Since the derivative is with respect to x, not [itex]\epsilon[/itex], we can write [itex]\phi(x, \epsilon)'= f(x, y, \epsilon)[/itex] and set [itex]\epsilon= 0[/itex] in that equation:
[itex]\phi(x, 0)'= f(x, y, 0)[/itex].
 

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