How can you determine y's behavior from a direction field graph?

[dwilmer]

How can you determine y's behavior from a direction field graph??

lets say i have the equation y' = 3 + 2y and i make a direction field graph.

I find equilibrium position at y = -3/2.

If question asks me to determine behavior of y as t approaches infinity, isn't this impossible, without solving the equation?

If i follow along the graph so that t is really big, then the graph still looks the same: a horizontal line at y = -3/2, where the slope of graph is zero. So how can i infer anything about y's behavior without first solving for y?

 

[HallsofIvy]

No, the graph is NOT "a horizontal line at y= -3/2". That line is a part of the direction field graph and you can conclude from it that "if y= -3/2 for any t then it will remain -3/2 for all t". But what if the initial value of y is NOT -3/2?

What does the direction field graph look like for y> -3/2? If y(0)> -3/2, what happens to y(t)?

What does the direction field look like for y< -3/2? If y(0)< -3/2, what happens to y(t)?