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.