Differential Equations Fall 2016
Comments on the first exam

Scores for the first exam have been posted on blackboard.  Exams will be returned in class on Friday, 9/30/16.  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 are shown in the histogram below.

exam 1 histogram

The histogram shows that most of the class did very well, with nearly 2/3 of the scores are at 90 or above.  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. 

Partial Credit System.    I assign partial credit on each part of each problem by thinking about how I would assign a letter grade to that particular part.  If I think an answer deserves an A I assign be 90% and 100% of the points; if I think it deserves a B
I assign between 80% and 90% of the points.  So if you got less than 70% of the points, that means I thought that particular answer was unsatisfactory.  If you got less than 60% that means I thought that particular answer was worthy of a failing grade.  This tends to make my partial credit assignments somewhat more generous than many other math teachers.  On the other hand, I rarely curve an exam.  If you got a total for this exam in the 90's, then your overall grade for the exam is in the A range.

Portfolios.  If you handed in a portfolio for review, please look at the review sheet I put in the front.  If I requested any changes to what you have put together, please make those changes as soon as possible, and before the next exam.  The review sheet must stay in the very front of your portfolio.  If you wanted to have your portfolio reviewed but forgot to bring it to the exam, you may bring it to my office hours by Thursday, 10/6 and I will review it there while you wait.

Comments About Specific Exam Questions

Problem 1.   I counted two kinds of errors:  circling a term that does not apply, or failing to circle one that does apply.  For each part A, B, and C, I indicated the number of errors like this: 2X means two errors.  But I did not specifically indicate where the errors are -- you will have to figure that part out.

Problem 2.  The class as a whole did not do very well on these discussion items.  This indicates to me that most students are missing some key concepts about differential equations.  I will summarize key points below

2A.  Here is the key idea of an autonomous model.  If you perform two experiments with the same initial conditions but with different starting times, do you observe the same behavior of the solution curve?  If so, then the model is authonomous.  Matheamtatically, this means that the later solution curve will appear to be the same as the earlier solution curve, but shifted horizontally consistent with the later starting time.  For algae growing in a pond, we would NOT expect this to hold.  Algae need sunlight to grow, so if you started two experiments with the same initial conditions but starting one at sunset and the other at sunrise, you would predict quite different population curves to result.  So we would not expect an autonomous model to be valid in this case.

2B.  The key ideas are as follows.

1.       A solution going to infinity in finite time means that on a particular solution curve there is a vertical asymptote  for some t*, or in other words, on that curve y goes to infinity as t approaches t*.

2.      The practical significance is that from an initial state, the model only allows us to predict future evolution for limited amount of time.  In the case of the example, for any initial point (say (160,29)) there is a unique solution curve (y = tan(t + C) where C = arctan(29)-160).   We can predict what y will be for a short amount of time only.  That is, given that y = 29 at time 160, we can predict what y will be at time 161, but not at time 163.  For a model of a real system, this might indicate a flaw in the model (if we have reason to believe the system will continue for extended periods of time, or that physical constraints impose a limitation on the size of the y variable) but it might also predict some catastrophic event in the future of the model.  In a population model, for example, it is possible for the model to predict a population that reaches 0 in finite time.  In practical terms, that means the population has died out.  The model might predict that the solution curve continues into negative P values beyond that point, or even goes to negative infinity in finite time.  These don’t necessarily invalidate the model – we know that the only relevant part of the model is where it predicts positive population values.

3.      The final question asks whether there are any conditions that tells us that solution curves do not go to infinity in finite time.  This is not about the specific example given, but for differential equations in general.  If you are given a differential equation  and an initial condition, is there a way to predict whether the solution curve might go to infinity in finite time?  Here is one answer: for an autonomous equation, look at the phase line.  If the conditions of the existence and uniqueness theorems hold, and if the initial y value is lower than an equilibrium solution, then the entire solution curve must be below that equilibrium value, and therefore y cannot go to infinity.

2C.  The key ideas are as follows.

1.       The student’s conclusions are exactly the sort we would derive from a qualitative analysis and phase line for the given problem.

2.      And yet, if you solve the equation (it is separable) you will find that the conclusions are incorrect: solution curves that begin above the t axis do not approach that axis as an asymptote.

3.      How can this be?  The answer is this: our qualitative analysis methods are valid when uniqueness holds.  (Why is that?  How do we use uniqueness to derive conclusions about asymptotes?)

4.      A complete correct answer for your exam solutions should explain where and why the uniqueness theorem does not apply for this problem, and also explain specifically why this failure invalidates our usual conclusions about asymptotes.  It should also find the solution curves for the given differential equation, and show directly that the conclusion about asymptotes is incorrect.

 4.  This problem asks for ALL possible solutions.  The method of separation of variables does NOT find all the possible solutions, because you have to divide both sides of the differential equation by something that might equal 0 for some value of t.  Division by zero in this situation doesn't mean that there can be no solution, but you cannot find it by the separation of variables technique.  For your exam solutions, you should indicate exactly the circumstances that make separation of variables invalid, and check by some other method whether any solution curves exist in those circumstances.