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Show that x(t) approaches infinity in finite time

  1. Jul 26, 2011 #1
    1. The problem statement, all variables and given/known data

    Consider the equation [tex]\dot{x} = rx + x^3,[/tex] where [tex]r>0[/tex] is fixed. Show that [tex]x(t) \rightarrow \pm \infty[/tex] in finite time, starting from any initial condition [tex]x_{0} \neq 0.[/tex]

    2. Relevant equations

    I can think of none.

    3. The attempt at a solution

    The idea alone of x(t) approaching infinity in "finite time" throws me for a loop. Does this merely mean that |x(t)| becomes very large very quickly? That it is increasing (x > 0) or decreasing (x < 0) increasingly quickly as it "moves away from the origin"? It strikes me as impossible for something to "become infinite" in a finite amount of time. This is from Strogatz' "Nonlinear Dynamics and Chaos", Exercise 2.5.3 if you are interested.

    Thank you!
     
  2. jcsd
  3. Jul 26, 2011 #2

    hunt_mat

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    Solve the equation!! You have:
    [tex]
    \frac{1}{x(r+x^{2})}\frac{dx}{dt}=1
    [/tex]
    Use partial fractions to separate to get:
    [tex]
    \frac{1}{x(r+x^{2})}=\frac{1}{rx}-\frac{1}{r(r+x^{2})}
    [/tex]
    And then simply integrate, this will give the answer.
     
  4. Jul 26, 2011 #3
    I would assume the phrase "approaches infinity in finite time" means the graph has an asymptote.
     
  5. Jul 26, 2011 #4

    hunt_mat

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    Indeed. There is a vertical asymptote as a certain finite value of t.
     
  6. Jul 26, 2011 #5
    I think I now understand the math behind this, although I'm unable to come up with a physical situation that models it?
     
  7. Jul 26, 2011 #6

    hunt_mat

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    First off, can you do the integral?
    [tex]
    \frac{1}{r}\int \frac{1}{x}-\frac{1}{r+x^{2}}dx=\int 1dt
    [/tex]
    Physically this would represent a shock wave of some kind.
     
  8. Jul 27, 2011 #7
    We end up with: [tex]\frac{ln(x) - (\arctan(x/\sqrt{r})/\sqrt{r})}{4} = t.[/tex] So if this is a shockwave, what exactly is x(t) modeling? Surely not position?

    Sorry for what I can kind of tell are pretty basic questions. I've had very little experience with differential equations so I'm sort of learning as I go along in Strogatz.
     
    Last edited: Jul 27, 2011
  9. Jul 27, 2011 #8

    hunt_mat

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    You also have a constant of integration in there somewhere which you can calculate.

    x would represent something like the gradient of the wave at a given time.

    So what values on the RHS would make it infinite? look at both the log and arctan terms.
     
  10. Jul 28, 2011 #9
    Are you sure about that partial fraction simplification? I get
    [tex]\frac{1}{rx} - \frac{x}{r(r+x^2)}[/tex]
     
  11. Jul 28, 2011 #10

    hunt_mat

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    I think you could be right there...
     
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