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Uniqueness of a solution

  1. Sep 6, 2011 #1
    1. The problem statement, all variables and given/known data
    Show that the solution of dx/dt=f(x), x(0)=xo, f in C^1(R), is unique


    2. Relevant equations
    C^1(R) is the set of all functions whose first derivative is continous.
    F(x)=integral from xo and x (dy/f(y))

    3. The attempt at a solution

    Assume phi1(x) and phi2(x) are both soultions. Then d(phi1(x))/dt=f(phi1(x)) and d(phi2(x))/dt=f(phi2(x)). Consider phi1(x)-phi2(x). d(phi1(x)-phi2(x))/dt= f(phi1(x))-f(phi2(x))...
    I need to prove the two solutions are infact equal. Also it says in my book that every solution x(t) must satisfy F(x(t))=t, with phi(t)=F^-1 (t) (F Inverse)
     
  2. jcsd
  3. Sep 7, 2011 #2

    lanedance

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    Homework Helper

    shouldn't the two solutions be functions of t?
     
  4. Sep 11, 2011 #3
    yes. I am sorry. I made a mistake. the two solutions hsould be phi1(t) and phi2(t).
     
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