ODE with parameter question

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  • #1
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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?
 

Answers and Replies

  • #2
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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.
 
  • #3
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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.
 
  • #4
HallsofIvy
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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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