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Domain of solution to Cauchy prob.

  1. May 4, 2010 #1
    Prove that the solution of the CP

    [tex] y'=-(x+1)y^2+x [/tex]
    [tex] y(-1)=1 [/tex]

    is globally defined on all of [tex] \mathbb{R} [/tex]

    How would you go about this? I thought about studying the sign of the right member if the equation. But what would I do next?
     
  2. jcsd
  3. May 4, 2010 #2

    HallsofIvy

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    What are you allowed to use? The fact that both [itex]f(x,y)= -(x+1)y^2+ x[/itex] and f_y(x,y)= -2(x+1)y[/math] are continuous for all (x, y) and the fundamental existence-uniqueness theorem should do it.
     
  4. May 4, 2010 #3
    Hi HallsofIvy.
    I guess I'm only allowed to use the theorem of local existence and uniqueness and the fact that (f being locally lipschitz and continuous) if for any compact K there exists a constant [math] C [/math] such that
    [tex] |f(x,y)| \leq C(1+|y|) \forall x \in K[/tex] and [tex]y \in \mathbb{R}^n [/tex]
    then there is a global solution, otherwise this exercise wouldn't be marked as "pretty hard" in my book.
     
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