3 we would like to use euler s method to approximate the solution to t
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.
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