# Slope Fields and Determing Behavior of Any Solution

## 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.

## 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?

Last edited:

HallsofIvy