3. We would like to use Euler's method to approximate the solution to the following initial value problem:

dy = 4(y-4)²(y-1), y(0) = 3dt

(a) To get an idea of what the solution should look like, draw a slope field for this equation and identify any equilibria.

For each equilibrium, is it stable, unstable, or neither?

(b) On the same axes, sketch the solution starting at y(0) = 3. What is the long term behavior of the solution?

(c) Use Euler's method with At = 0.2 to approximate y(t) up until t = 0.6. Graph your approximate solution. Does

this match the behavior predicted by the slope field? Answer in a complete sentence.

(d) The phenomenon in part (c) is known as "numerical instability" It turns out that Euler's method can be numerically

unstable if we don't choose At well. Find a value of At that will give a more reasonable approximation for y(0.6).

Use the new At to create a new approximate solution until t = 0.6. Graph your new approximate solution.