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.

Fig: 1