Approximate grade ranges (for guidance only):

44-50 A
38-43 B
32-37 C
26-31 D
0-25 F

Wednesday, December 4, 7:00-8:15pm (evening) in room 66 Library (enter from the Wright Street side, down the steps to the right)

Bring your I-card to the exam. No I-card = no exam.

Students with a DRES accommodation should arrange to take their exam at DRES, on Thursday, December 5.

Conflict exam – Thursday, December 5, 8:00-9:15am (morning). You can request here to take the conflict exam, but read the next paragraph before you do so:

• The conflict exam will be given only if you present written evidence that you will miss (or did miss) an exam for a legitimate reason. Religious time conflicts, university-related sports competitions, family emergencies and illness are examples of legitimate reasons for an absence. If a conflict can be predicted in advance, then the evidence should be presented at least one week prior to the scheduled exam date. In case of religious observances, complete this form as soon as possible.

• In case of illness, an absence letter from the Dean of Students will suffice. 
• Travel and leisure plans, even for family events, are never a legitimate reason for missing an exam.

Exam Coverage

The midterm is designed as a 1-hour exam, with some multiple-choice and some short-answer problems. You will have 1 hour and 15 minutes to work on the exam. The exam includes a formula sheet, exactly as on the Practice Problems (see below). 

Material: Lecture 25 (Monday, October 28) on “Homogeneous Linear Systems with Constant Coefficients” through Lecture 35 (Monday, December 2) on “Conserved and Dissipated Energies, cont. See the handouts page for detailed class material and coverage. 

All Webassign Homework and Written Homework on this material is examinable. 

How to study

Active study is the key to success. Write summary notes of the important concepts and methods, based on your lecture notes and class handouts on each section (solutions are posted on the handouts page). Passive study, such as flipping through your notes, will not prepare you for the exam.

The test could cover:

• modeling
• solving
• graphing
• interpreting

To help with “solving” DEs, make a checklist of the main types of DEs/systems we have covered, and the method used to solve each type.
Practice looking at the form of a DE and do not fixate on the variable letters e.g. d²x/dt² = -a²x is the same DE as d²y/dt² = -a²y.
Express the methods as algorithms or checklists: step 1, step 2, and so on, so that you have a plan of action for each type of problem.
After solving a DE or system of DEs, always check your work by “verifying” – plug your solution into the DE to check that it works.

Memorize the solutions of the following differential equations, and practice writing down the solution until you can do it without thinking:

• dy/dt – ay=0      or dy/dt = ay               (solution y=Ce^at)
• d²y/dt² – a²y=0 or d²y/dt² = a²y     (solution y=c₁e^at+c₂e^-at)
• d²y/dt² + a²y=0 or d²y/dt² = -a²y   (solution y=c₁cos(at)+c₂sin(at))

Memorize the:

• trace-determinant diagram (download here), and practice drawing it

Test yourself on the CONCEPTS of the course, by asking yourself questions:

• what is the characteristic equation of a homogeneous linear system of DEs? 
• for what kind of system of DEs do we solve using eigenvalues and eigenvectors?
• why does that solution formula (in terms of eigenvalues and eigenvectors) solve the system? can you verify it? 
• how do you get a real-valued solution to the system when the eigenvalues are complex? DO NOT memorize the formula.  Instead, know the method from Lecture 26.
• what goes wrong with the general solution when the eigenvalue is real and repeated?  
• how do you use the trace-determinant diagram? (download it here, and practice drawing it)
• for each region in the trace-determinant diagram, state the type of phase portrait it corresponds to, draw a typical phase portrait, and say whether the equilibrium point at the origin is stable or unstable for that type of phase portrait
• how is the matrix exponential defined? 
• is the law of exponents always valid for matrix exponentials? if not, then under what circumstances is it valid?
• how do you compute the exponential of a diagonal matrix? of a conjugated matrix? of a diagonalizable matrix?
• how do you linearize a nonlinear system, around an equilibrium point? (there is a general method that works for any equilibrium point; and if the equilibrium is at the origin then there is a simpler method)
• what is a “generic” phase portrait?
• what does the Linearization Theorem tell you?
• what kinds of long-time behavior have you seen for linear systems in 2 dimensions? in 3 dimensions?
• what kinds of long-time behavior have you seen for nonlinear systems in 2-dimensions?
• can you give an example of a nonlinear system with 4 equilibrium points? with infinitely many equilibrium points?
• can you sketch typical trajectories for the SI model? (That is, the SIR system except looking only at trajectories in the is-plane.) Which point on the trajectory corresponds to the start of the epidemic, and which point corresponds to the end of the epidemic?
• if you are given an energy formula, how would you check whether the energy is conserved? dissipated?

Then work lots of problems: 

• Re-work all homework problems – make sure you know how to do every problem on Homeworks 8, 9, 10.
• Ask for help at an office hour or collaborative homework session on every problem you are not sure of.
• Attempt the Practice Problems (with answers here for the problems “Review on homogeneous systems”). Answers for most other problems are given at the end of the textbook, on pages 589-635. (Note: the file also includes the formula sheet that will be given on the exam.)  
• Once you are confident on the fundamentals, assign yourself “backwards” challenges such as “Find an example of a 2×2 matrix whose phase portrait is a spiral source.”
• Some extra practice problems have been posted on Webassign.

Get help

• Collaborative Homework Sessions (as usual): Monday, 4-5pm in 243 Altgeld Hall
  Wednesday 4-5pm in 243 Altgeld Hall
  Thursday 5-6pm in 443 Altgeld Hall
• Office hours – Tuesday, December 3, 4-5pm in 376 Altgeld Hall

Exam rules

You must not communicate with other students or anyone else except the proctors, during the exam. This includes the turn-in period after the exam.
No written materials or electronic materials of any kind are allowed.
No phones, calculators, iPods or electronic devices of any kind are allowed for ANY reason, including checking the time. You may use a simple wristwatch, but not a smartwatch.
Put all electronic devices in your bag before the exam begins; they must not be on your person.
Cheating of any kind or violations of exam rules will be treated extremely seriously.

Regrade requests

All requests for regrading must be received by 12pm on Friday, December 13. No exceptions other than for illness or other excuse that prevents you from coming to campus. 
If you believe that there is a mistake in the grading or scoring, then write a BRIEF (one-sentence) explanation on the cover page of your exam, and give the exam to Professor Laugesen by 12pm on that day.