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Time to Travel to One Infinity and Back?

  1. Jan 18, 2012 #1
    1. The problem statement, all variables and given/known data

    Consider the ODE x' = x2 + ε, where ε is a small number. Find the time T = T(ε) it takes for the solution to travel from x = -∞ to x = ∞.

    Let T1 = T1(ε) be the time it takes the particle to travel to x = -1 to x = 1. Show that T/T1 → 1 as ε → 1.

    2. Relevant equations

    Uhmm... we haven't really learned any yet. This is qualitative analysis of ODEs.


    3. The attempt at a solution

    I have no idea where to start really. I thought that to find time, we could treat x' as a velocity, and find the position function by integrating x'. Once we had this position function, I thought it would be okay to divide position by velocity to find time... but I don't even know what that would represent! From negative infinity to infinity, no less...
     
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  3. Jan 18, 2012 #2

    Dick

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    In this case you have to solve the ODE before you start making statements about the qualitative behavior. Start with the second part. Solve the ODE with initial conditions x=(-1) at t=0 and figure out how long it takes you to get to x=1.
     
  4. Jan 18, 2012 #3
    Are the initial conditions arbitrarily set? As in it's just convenient to use -1 at t=0 because... well I don't know... :/

    I mean, how exactly did you come to the conclusion that that was the best course of action to start with?
     
  5. Jan 18, 2012 #4

    Dick

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    I think it's the best course of action because I don't see how else you would do it. And no, the initial conditions aren't particularly arbitrary. If x(0)=(-1) then if you can find the value of t such that x(t)=1 (or at least the limit of that as epsilon goes to 0) then that value of t is the time required to go from x=(-1) to x=1.
     
  6. Jan 19, 2012 #5
    So I am trying to solve it now, but I am a little put off by the ε. I mean, I understand how to solve an ODE with just x' = x2... but when you say the limit of ε → 0, is that a final step or something? I'm just having trouble getting this problem off the ground.

    EDIT: I'd like to add that my professor "kind of" went over this question today. He stated that if ε > 0, then there would be no equilibrium solutions, but if ε < 0, then there are two equilibrium solutions on a "bifurcation line" thing. I remember the term "saddle node bifurcation" and that if ε < 0, then the two equilibrium solutions are -√ε and √ε. I feel that I should use this information to solve this question... but to be honest I didn't really understand any of this bifurcation stuff.
     
    Last edited: Jan 19, 2012
  7. Jan 19, 2012 #6

    Dick

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    You really need to be able to solve differential equations to understand them. Forget all the other stuff for now. Just concentrate on that. Put ε=1. Try and solve x'=x^2+1.
     
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