Differential Equations Fall 2016
Comments on the second exam

Scores for the first exam have been posted on blackboard.  Exams will be returned in class on Tuesday, 11/7/16.  It is important that you come to class to pick up your exam because you are assigned to work with a group to produce a complete set of correct solutions that will be due Friday, 10/7.  Exam solutions are a required component for the course portfolio.  Specific directions are given here

When you get your exam, please double check that I have added up the points correctly and also that the score on your paper is the same as the one posted on blackboard.  I try hard not to make errors totaling or recording scores.  But if an error has been made, I will want to correct it as soon as possible.

The exam scores  and each student's average for exams 1&2 are shown in the histograms below.

exam 2 histogram

The first histogram shows that most of the class did very well, though the curve was not as high as on the first exam.  This time 2/3 of the scores are at 85 or above.  The second histogram shows that the at this point everyone is earning a satisfactory grade, and almost everyone is on track for a final grade of B or better. 

Anyone who is unhappy with his or her performance should have a discussion with me.  I may be able to suggest ways to study more effectively. 

Specific Comments on Exam Problems.   These comments may be helpful in creating your exam solutions.

1.a.  Your answer to this should say specifically how the two graphs under consideration are created.  In each one, what gets plotted, where, and how do you calculate that?

1.b.  Your answer should :
  1. Say what an equilibrium point IS.  (ie, define equilibrium point)
  2. Say what stable means
  3. Say what unstable means
  4. Use language carefully.  A solution curve can approach a point, but a particular point or a particular initial value doesn't approach anything -- it is a static object.
2.  A complete solution must include ALL solutions.  That means you have to include the arbitrary constants that arise in integration steps.  It also means you have to be aware of the possibility of missing solutions.  For example, if you divide both sides of an equation by x, that excludes the possibility that x = 0.  So you have to check that as a separate possibility.  Likewise y.

3.a.1. 
Make clear what an equilibrium point IS in the context of this problem, and why it is possible or impossible for more than one to exist.

5.c.1.  A correct diagram should show representative solution curves with correct limiting behavior, meaning correct tangency conditions as t goes to infiinity.

6.  Some students used the theorem on page 319 to answer parts a and b.  Some students used the exponential matrix approach from the handout.  Both are valid, but the theorem on 319 is faster and less work.  The point of the handout is to unify all the different cases in a simple and meaningful way, but that doesn't necessarily mean that it is computationally easier in any particular case.  For part c your description should include something specific about the shapes of the solution curves -- just saying they go to the origin is not sufficient.  See page 320.

7.c.  In this problem you are given a proposed solution.  It is specified as Y(t) = eAtY0.  Note that this is a matrix (eAt) multiplied by a vector (Y0) and so is itself a vector.  You are asked to verify that this is a solution to the IVP.  That means you must show that this function satisfies the differential equation (so you have to show its derivative is the same as what you get if you multiply it by A) and also that it has the specified value when t = 0.