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
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.
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
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.
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 :
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.
- Say what an equilibrium point IS. (ie, define equilibrium
- Say what stable means
- Say what unstable means
- 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.
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.
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
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.