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Autonomous DE Question

  • #1

Homework Statement


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]


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.
 

Answers and Replies

  • #2
gabbagabbahey
Homework Helper
Gold Member
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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]?
 
  • #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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