Slope Fields and Determing Behavior of Any Solution

[Bashyboy]

Homework Statement

The differential equation is y' = y^2
Draw a direction field for the given differential equation.
Based on the direction field, determine the behavior of y as t →∞. If this behavior depends
on the initial value of y at t =0, describe this dependency.

Homework Equations

The Attempt at a Solution

Okay, so I plotted the slope field by evaluating the derivative at several different y-values. This is what I observed:

The solution curve certainly depends on the initial y-value when t=0. If particular solution y(t) to the DE has a solution of the form (t=0, y>0) (passes through a point), then as t----> infinity,
y(t)----> infinity. On the other hand, if some particular solution y(t) passes through a point of the form (t=0, y<0), then as t--->infinity, y(t)----> 0

Are these correct observations; and have I used terminology and notation correctly?

Also, when considering solutions that pass through the negative portion of the y-axis, will some approach y=0 more quickly than others?

 

[HallsofIvy]

I am confused as to why you would say "On the other hand" for the initial value of y< 0 then say that its behavior is exactly the same as for y> 0.

And you seem to have left out an important special case- what if y= 0 for some t? And if y goes to infinity even if y is negative, how does it pass y= 0?

 

[Bashyboy]

What you have pointed out, I believe I have fixed.

In the case that some solution passes through (0,0), then as t---> infinity, y will remain zero, right?