Approximate grade ranges (for guidance only):
44-50 A
39-43 B
34-38 C
28-33 D
0-27 F
Wednesday October 2, 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, October 3.
Conflict exam – Thursday, October 3, 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 will be 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.
Material: everything covered in class through Wednesday, September 25, including Sections 1.1-1.3, 2.1-2.5, 2.7, 3.1-3.3 and all Webassign Homework and Written Homework relating to that material. Note that the modeling problems in Section 2.3 (Examples 1,3,4 on mixing, time-varying input concentration, escape velocity) were covered in class while doing other sections.
You do not need to learn topics that were not covered in class or homework.
How to study
Active study is the key to success. Write summary notes of the important ideas and methods, based on your lecture notes and class handouts on each section (handouts with solutions are posted on the handouts page). Passive study, such as flipping through your notes, will not prepare you for the exam.
The test will cover all four activities:
- modeling
- solving
- graphing
- interpreting
Solving. To help with “solving” DEs, make a checklist of the main types of DE we have covered (1st order constant coefficient linear, 1st order linear with variable coefficients, separable, Bernoulli, 2nd order constant coefficient linear), and the method used to solve each type.
Do problems 1-32 and 36-47 at the end of Chapter 2 on pages 133-136 (ignore problems 3,16,21,31 because they use the method of exact equations, which we did not cover; answers for the other problems are given on page 747). Practicing these problems will help you recognize the different types of DE.
Practice looking at the form of a DE and do not fixate on the variable letters e.g. dx/dt = ax is the same DE as dy/dx = ay, but is different from dx/dt=at.
Check your skill at recognizing the different types of DE by working these Problems. If needed, consult the Hints. When you are ready, check the Answers.
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 automatically:
Memorize the statements of the following theorems (you do not need to learn the proofs):
- Existence and Uniqueness Theorem for 1st order linear DEs, as stated on the first handout for Section 2.4 (second side of the handout); draw a picture to illustrate, like we did in class
- Existence and Uniqueness Theorem for 1st order nonlinear DEs, as stated on the second handout for Section 2.4; draw a picture to illustrate, like we did in class
- Existence and Uniqueness Theorem for 2nd order linear DEs, as stated on the handout for Section 3.2
- General solution Theorem for 2nd order linear homogeneous, as stated on the handout for Section 3.2
Test yourself on the CONCEPTS of the course, by asking yourself questions:
- what is a direction field for a 1st order DE? how do you draw it by hand? how do you use the direction field to help sketch solutions of the DE?
- what does “terminal velocity” mean? what does “extinction time” mean? what is the DE for Newton’s Law of Cooling?
- what is the “order” of a DE? what is the difference between an ODE and a PDE? what is the form of a 1st order linear ODE? a 2nd order linear ODE? what is an example of a 2nd order nonlinear ODE? what are initial conditions used for?
- what is an “integrating factor”, and when should you use it?
- why must you check first for constant solutions, when using the separable method?
- what does “finite-time blow-up” mean? what is an example of a nonlinear DE whose solutions blow up in finite time?
- can a 1st order linear DE with continuous coefficients (in standard form) have two solutions satisfying the same initial condition?
- what is an example of a 1st order nonlinear DE that has two solutions satisfying the same initial condition?
- what is an autonomous DE? what is a critical point? what is an equilibrium solution? what is an example of an autonomous DE that has an unstable equilibrium at y=3 and an asymptotically stable equilibrium at y=-1?
- what is the Euler update formula? how do you implement it graphically, on a direction field?
- what form does a 2nd order linear DE have? give an example with constant coefficients, and an example with variable coefficients
- what does it mean for a linear DE to be homogeneous? nonhomogeneous?
- what does it mean for a linear DE to have constant coefficients?
- what does the Superposition Principle say for linear homogeneous DEs?
- what does it mean to say that some formula for y(t) gives the “general solution” of a linear DE?
- what does it mean for two functions to be linearly independent?
- for what kind of DE do we solve using y=ert?
- what is the characteristic equation?
- when the roots of the characteristic equation are real and distinct, what solutions do they give for the DE?
- when the roots of the characteristic equation are complex numbers, what solutions do they give for the DE?
Then work lots of problems:
- Re-work all homework problems.
- Work new problems e.g. end of Chapter 2, Miscellaneous Problems, and problems from Sections 3.1-3.3 that are similar to things we did in class or on Webassign. Answers are at the back of the book.
- Attempt the Practice Exam (which includes the formula sheet that will be provided on the Midterm). Here are the Solutions to the Practice Exam. Here are the Webassign Practice Problems.
Get help to resolve your questions
- Collaborative homework sessions
- Monday 4-5pm in 243 Altgeld Hall
- Wednesday 4-5pm in 243 Altgeld Hall
- Thursday 5-6pm in 443 Altgeld Hall
- Regular office hour – canceled on Friday September 20 and 27, due to Professor Laugesen’s travel.
- Extra office hour – Tuesday October 1, from 4:00-5:00pm, in 376 Altgeld Hall
- Review day – in class, Wednesday October 2 – bring your questions!
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, October 11. 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.