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One dimensional mechanical system.

  1. Nov 7, 2009 #1
    1. The problem statement, all variables and given/known data

    Given the dynamical system [tex]\dot{x}=1-x^2[/tex], show that

    [tex]F(x,t)=\frac{1+x}{1-x}e^{-2t}[/tex]

    is a constant of that system, and obtain the general solution of the differential equation with [tex]F(x,t)[/tex]

    2. Relevant equations

    Above

    3. The attempt at a solution

    As [tex]F(x,t)[/tex] is a constant, then should satisfy

    [tex]dF=\frac{\partial F}{\partial x}dx+\frac{\partial F}{\partial t}dt=0[/tex]

    [tex]\frac{\partial F}{\partial x}=\frac{2(e^{-2t})}{(1-x)^2}[/tex]

    [tex]\frac{\partial F}{\partial t}=\frac{(1+x)(-2 e^{-2t})}{1-x}[/tex]

    Now, as [tex]\dot{x}=1-x^2[/tex]

    [tex]\frac{dx}{dt}=\frac{(1+x)e^{-2t}}{1-x} \frac{(1-x)^2}{e^{-2t}}=1-x^2[/tex]

    wich completes the proof.

    Now, to compute the general solution [tex]x(t)[/tex] of the problem, should I use the fact that

    [tex]\int\frac{\partial F}{\partial x}dx=-\int\frac{\partial F}{\partial t}dt[/tex]

    and use that

    [tex]\frac{\partial F}{\partial x} \frac{\partial x}{\partial F}=1[/tex]

    to find an integral for [tex]x(t)[/tex].

    Any kind of help is appreciated :D
     
  2. jcsd
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