Math 285 Test 2

Math 285 Test 2, Spring 2010

Friday, April 2, in class, worth 20%.
You may not use books, notes, or electronic devices on the test.

Material
Sections 3.1-3.6 plus boundary value problems, quizzes and homeworks. Ignore the material on Wronskians in Sections 3.1 and 3.2. (In class we covered linear independence of exponential functions and sine and cosine, without using Wronskians. You should do those problems on the Section 3.2 Summary handout). Ignore Undetermined Coefficients Rule 1 in the textbook. Instead use Rule 1 in the Undetermined Coefficients handout.

How to study Make summary notes of the important ideas and methods, from the lecture notes on each section. Pay attention to all four types of work:

modeling (e.g. know the physical meaning of m,c,k in Section 3.4)
solving
graphing (e.g. know how to graph x_c for undamped, underdamped, critically damped or overdamped oscillations, and the response x_p to forcing)
interpreting

Make a summary table of the main solutions and conclusions and graphs for mechanical vibrations, in Sections 3.4 and 3.6. There are four cases: undamped unforced, damped unforced, undamped forced, damped forced. We always assume the forcing function is periodic. Work through for yourself all the derivations of these solution formulas. In particular, the derivation in Section 3.6 of x_p(t) for the undamped forced case (nonresonant and resonant subcases) and in the damped forced case are examinable.

Write a brief paragraph: what is resonance, for undamped forced oscillations? what is practical resonance, for damped forced oscillations?

In the methods of Undetermined Coefficients and Variation of Parameters, we first find y_c and then find y_p, then apply the initial conditions to evaluate the constants in yc. Ask yourself: when should you use Undetermined Coefficients, and when use Variation of Parameters?

Learn to look at the form of a DE and not get fixated on the variable letters e.g. d²y/dx²+w²y=0 is the same DE as d²x/dt²+w²x=0. For any new DE, first ask yourself: is it homogeneous? is it linear? does it have constant coefficients? Then decide which method to apply.

Memorize three types of boundary value problems: Dirichlet, Neumann and periodic. Study the method of finding eigenvalues and eigenfunctions. Specifically, always consider three cases: positive, zero and negative lambda. Memorize Fredholm alternative and be prepared to use it to show that certain nonhomogeneous boundary value problems have(do not have) unique solutions.

Re-work all homework problems, and quiz preparation problems. Ask for help at an office hour or tutoring room on every problem you are not sure of. Then work new problems.

Attempt the Practice Test (on website, with solutions).