Autonomous DE Question: Solving for the Limit of y(t) | Initial Value Problem

[elvishatcher]

Homework Statement

Suppose that y(t) is a solution of the initial value problem
dy/dt = y(1-.0005y) , y(0) = 1
What is the limit \lim_{t\to\infty}y(t)

The Attempt at a Solution

If I try just separating it to solve for y(t) then I get \int{{dy}/{(y-.0005y^2)}} = x+C 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.

 

[gabbagabbahey]

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

Instead, look at where y(t) is increasing/decreasing and what turning points it has. Start by looking at t=0 - 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 t \to \infty?

 

[elvishatcher]

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 going to change anymore. Thanks.