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Autonomous DE Question

  1. Aug 18, 2012 #1
    1. The problem statement, all variables and given/known data
    Suppose that [tex] y(t) [/tex] is a solution of the initial value problem
    [tex] dy/dt = y(1-.0005y) , y(0) = 1 [/tex]
    What is the limit [tex] \lim_{t\to\infty}y(t) [/tex]


    3. The attempt at a solution

    If I try just separating it to solve for [tex]y(t)[/tex] then I get [tex]\int{{dy}/{(y-.0005y^2)}} = x+C[/tex] which I can't figure out how to solve. I'm at a loss as to what else to do. Is there some way I should be able to predict the behaviour of the function with just the initial value and the DE, or am I missing something about how to evaluate the integral or solve the DE? I tried predicting what the function would do based on what I was given, but I got the wrong answer and figured it was because I was doing it based on the y values, but not knowing the actual function I didn't know which y values would occur as t went to infinity.
     
  2. jcsd
  3. Aug 18, 2012 #2

    gabbagabbahey

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    Homework Helper
    Gold Member

    You can solve the DE by using partial fraction decomposition, but that isn't really necessary here.

    Instead, look at where [itex]y(t)[/itex] is increasing/decreasing and what turning points it has. Start by looking at [itex]t=0[/itex] - is the function increasing or decreasing there? A small time later, is the function increasing or decreasing? Does it eventually reach some turning point and start increasing/deceasing towards some finite value as [itex]t \to \infty[/itex]?
     
  4. Aug 18, 2012 #3
    Ah, I see it now. I was getting a little mixed up with the y's and the t's, and I wasn't thinking about the fact that once it settles into an equilibrium value it's not gonna change anymore. Thanks.
     
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