Test #2 Information

General Information
**Wednesday, November 8, 9:00-9:55 in the usual classroom.
**The test will begin promptly at 9:00 and end promptly at 9:55.  If you arrive late, you may still take the test, but you must turn it in at 9:55.
**Please bring your I-card to the exam.
**No notes or books may be used on the test.
**No calculators may be used on the test.
**The exam covers Sections 3.1 (skipping Wronskian in 3.1 and 3.2), 3.2, 3.3, 3.4(skipping pendulum), 3.5, 3.6, 3.8(skip whirling string), Iode Project III, Homework Assignments #5, 6, 7, 8
**There will be several slightly different versions of the test to discourage any cheating.  I very much hope it won't be necessary, but if any cheating is detected, I will certainly follow through on imposing the maximum penalties allowed by the university.
**I may ask you to move your seat if it appears that you are copying or that someone is copying from you.

What will the test be like?
**If you know the material, you should be able to complete the test comfortably in 50 minutes or less.
**The test will include some very basic questions, some medium-difficulty problems, and one or two more challenging problems.
**The types of questions that may appear on the test include true/false, state the definition, give an example, explain a concept, do a proof (similar to those you've done for homework), calculational problems (similar to homework).
**Many of the problems will be quite similar to homework problems!  However I will avoid some of the very long computations which appeared on the homework and some of the difficult graphs which were time-consuming to do without a calculator.  You may be asked to outline the procedures, or to do just part of such a problem, or to do a problem of the same type in which the computations happen to be short.
**You will need to show your work on the test.
**Problems will be written in such a way that a calculator is not needed.

Definitions
Be sure you know the definitions of the following terms.  You should be able to state these definitions precisely, not necessarily with exactly the same words as the textbook, but with exactly the same mathematical meaning.  You should also be able to give examples.

1. nth-order linear differential equation
2. homogeneous, non-homogeneous (in the context of nth order linear differential equation)
3. linearly dependent functions, linearly independent functions (for a set of n functions)
4. complementary solution (also called complementary function)
5. characteristic equation
6. period
7. amplitude
8. frequency, circular frequency
9. phase angle, time lag
10. overdamped, critically damped, underdamped
11. natural frequency of an undamped mass-spring system
12. resonance and pure resonance for an undamped mass-spring system
13. transient solution
14. steady periodic solution
15. practical resonance for a damped mass-spring system
16. endpoint problem
17. eigenvalue (in the context of Section 3.8)
18. eigenfunction (in the context of Section 3.8)`

Theorems
You should be able to state the following theorems, understand what they mean, and be able to use them.  Unless otherwise stated, you do not need to be able to prove them.

Section 3.1, Theorems 1, 2, 4, 5, 6 (note - these are all for order 2 and are repeated in a more general form for order n in Section 3.2 and 3.3 so it would be okay to skip these as long as you know the theorems from 3.2 and 3.3)
Section 3.2, Theorems 1, 2, 4, 5 (5 is saying that the general solution is a particular solution plus the complementary solution)
Section 3.3, Theorems 1, 2, 3

You should be able to prove 3.2, Theorem 1 (try it for n=2 or 3) and Theorem 5

Review Problems
One of the very best ways to study for the test is to rework your old homework, especially problems that you missed or were unsure of.  Try to do them without looking at books or your notes - this is what you will be doing on the test!  Please let me know if you find any errors in the solutions I've posted.

However I will avoid some of the very long computations which appeared on the homework and some of the difficult graphs which were time-consuming to do without a calculator.  You may be asked to outline the procedures, or to do just part of such a problem, or to do a problem of the same type in which the computations happen to be short.

You should be able to:

1. Determine whether or not a given set of n solutions of an nth order linear homogeneous equation is linearly independent or linearly dependent.
2. Solve homogeneous linear equations with constant coefficients.  Find the general solution and solve the initial value problem (if initial conditions are also given).
3. Use undetermined coefficients to solve non-homogeneous linear equations with constant coefficients.
4. Use variation of parameters to solve non-homogeneous linear equations (I'll just ask order 2).
5. Set up and solve mass-spring problems, both with and without damping and with and without a forcing function.
6. For mass-spring problems, be able to find amplitude, frequency, phase angle, period, time lag, sketch of graph (if not too complicated!).
7. Describe what is meant by resonance, pure resonance, practical resonance.  Describe under what circumstances each will occur and what the graphs look like.
8. For an undamped mass-spring system, find the frequency which results in pure resonance.
9. For a damped mass-spring system, find the frequency which results in practical resonance (if any).
10. Find eigenvalues and eigenfunctions (as in Section 3.8).
11. Solve problems concerning the deflection of a uniform beam or the buckled rod.  I'll give you equations (20) and (26) from 3.8 so you don't have to memorize those.
12. The following trig identities will be given on the exam if needed, so you don't have to memorize them
    cos(b-a)=(cos a)(cos b) + (sin a)(sin b)
    2(sin A)(sin B) = cos (A-B) - cos (A+B).
13. This list gives the major topics but is not meant to be all-inclusive. Remember to study the definitions and theorems, too.

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