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Asymptotic behaviour of 1st order ODE

  1. Nov 20, 2009 #1
    I have a first order ODE
    [tex]
    yy'=a(x)+b(x)c(y)
    [/tex]
    and all I want to know is [itex]y'(\infty)[/itex]. Is there an easy way to find out or at least for some special forms of [itex]c(y)[/itex]?

    Eventually I'd like to find functions a, b, c such that there is a solution with [itex](x=\infty,y=-V)[/itex] [itex](x=\infty,y=V\alpha)[/itex] for any V where [itex]\alpha[/itex] is a given factor. Preferably with [itex]a(x)=-Ax^{-n}[/itex]
     
    Last edited: Nov 20, 2009
  2. jcsd
  3. Nov 22, 2009 #2

    MathematicalPhysicist

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    Gold Member

    If you know that [tex]y(\infty)=-V[/tex], so you know that [tex]y'(\infty)=[a(\infty)+b(\infty)c(-V)]/(-V)[/tex], you must know beforehand what are b,c,a are to find it.
     
  4. Nov 22, 2009 #3
    I can chose a, b, c as given. And V is basically whatever y' is in one solution.
    The important point is that the other y' solution at infinity should be a given factor [itex]\alpha[/itex] of the first solution.
    Maybe I should say where the problem came from. I basically want to find a physical law such that an object bouncing off a wall will lose a given part [itex]1-\alpha[/itex] of its velocity. So I have the wall force a and the damping force b which should both be concentrated at the wall only.
    And [itex]yy'[/itex] is the force on the particle after a mathematical transformation.

    Oh I just notice I might have mixed up the variables after the transformation...
    I want to know (x,y) as given with the points. I don't need y'.
     
    Last edited: Nov 22, 2009
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