Approximate grade ranges (for guidance only):

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

Wednesday, October 30, 7:00-8:15pm (evening) in room 1404 Siebel Center.

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, October 31.

Conflict exam – Thursday, October 31, 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 (NA) 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 Exam (see below). 

Class Material: Boyce & DiPrima Sections 3.3-3.8, 5.2; and material from Polking-Boggess-Arnold up through Lecture 24 (Friday, October 25) on “Linear Independence and General Solutions”.

All Webassign Homework and Written Homework are examinable. 

How to study

Study actively.  

Write summary notes of the important ideas and methods and models, based on your lecture notes and class handouts on each section. (Remember that handouts with solutions are posted on the handouts page.)

Passive study, such as flipping through your notes, will not prepare you for the exam. Just doing practice problems will not prepare you for the exam. 

The test will cover:

• modeling
• solving
• graphing
• interpreting

To help with “solving” DEs, make a checklist of the main types we have covered, and the method used to solve each type. How do you decide which method to use? This flowchart can help.
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, 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 statements of the:

• Euler’s formula e^ib=cos(b)+i sin(b)
• Rules 1 and 2 for Undetermined Coefficients
• Definition of an equilibrium point for an autonomous systems of DEs, and the Existence and Uniqueness Theorem for linear systems, as stated on the handout for Wednesday October 23 (8.3 and 8.4 Equilibrium Points, and Existence and Uniqueness)
• Statement of the General Solution Theorem for homogeneous linear system of n equations, as stated on the handout for Friday October 25 (8.5 Linear Independence and General Solutions)

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

• what goes wrong with our general solution to the DE if the characteristic equation has repeated roots? 
• how do you find a second solution of the DE in that case? 
• what is the method of Reduction of Order, and when should you use it?
• what is an Euler type DE? (special type of linear homogeneous DE – what does it look like?)
• on what type of DE should you use Undetermined Coefficients? what are the three Rules governing the method?
• why is the second Rule important?  
• on what type of DE should you use Variation of Parameters? what do you need to know before you can apply the method?
• what is an unforced (free) oscillator DE? what does it represent physically? 
• what is the natural frequency? what is its relation to undamped, unforced oscillators?
• what does “simple harmonic motion” refer to? what is the solution formula for simple harmonic motion?
• when is the oscillator underdamped? critically damped? overdamped? in each case, what is the solution formula and what does the solution graph look like? 
• what do quasi-frequency and quasi-period mean? in what situation do they arise?
• give an example of a linear DE that has a decaying solution, but for which 100% of randomly chosen solutions will grow exponentially. This situation sounds like a contradiction – explain why it is not.  
• what do amplitude, phase, frequency, and period mean? give an example and state what its amplitude, phase, frequency, and period are
• what is a forced oscillator DE? what does it represent physically? 
• what do forcing frequency and forced response mean? in what situation do they arise? 
• what does steady state solution mean? when is a solution called transient?
• for a forced undamped oscillator, does periodic forcing generate a periodic response? sketch a typical periodic response graph, and label the period on the graph; then sketch how the response amplitude depends on the forcing frequency. 
• what does “resonance” mean, and when does it occur?
• for a forced damped oscillator, does periodic forcing generate a periodic response? sketch a graph showing how does the response amplitude depends on the forcing frequency 
• what does “practical resonance” mean, and when does it occur? 
• for what type of DE should you try a series solution? what does a “recurrence relation” mean?
• what form does a linear homogeneous system of DEs have? a linear nonhomogeneous system?
• what is an example of a nonlinear autonomous system? a nonlinear nonautonomous system?
• is the origin an equilibrium point for every homogeneous linear autonomous system (i.e. homogeneous linear constant coefficient system)? 
• is the origin an equilibrium point for every nonlinear autonomous system?  
• what does it mean for two vectors to be linearly independent?
• how can you turn a 2nd order DE into a first order system?
• how do you draw a direction field for an autonomous first order system? or match it to a given system?

Then work on lots of problems: 

• Re-work all homework problems. Ask for help at an office hour or collaborative homework session on every problem you are not sure of.
• Work on new problems from the textbook. Answers for most problems are given at the end of the textbook, on pages 589-635.  
• Attempt the Practice Problems and note the formula sheet on the last pages. (Here are the Practice Problem Solutions.) Some extra practice problems are on Webassign.

Get help

• Collaborative Homework Sessions
  Monday 4-5pm in 243 Altgeld Hall
  Wednesday 4-5pm in 243 Altgeld Hall
  Thursday 5-6pm in 443 Altgeld Hall
• Office hours – Friday, October 25 2:30-3:30pm, and Tuesday, October 29 4:30-5:30pm, 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, November 8. 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.