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ODE with parameter question

  1. Sep 19, 2012 #1
    1. The problem statement, all variables and given/known data

    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?
     
  2. jcsd
  3. Sep 20, 2012 #2
    There is a theorem on the dependence of ODE solutions on parameters. I am sure it has been covered in your course of ODEs.
     
  4. Sep 20, 2012 #3
    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.
     
  5. Sep 20, 2012 #4

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    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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