## Math 285 Test 2, Spring 2010

Test 2 Solutions. Median score: 83/100

Approximate grade ranges (out of 100 points):

• A: 90-100
• B: 80-89
• C: 65-79
• D: 50-64
• F: <50

Wednesday 31 March, in class, worth 20%.
You may not use books, notes, or electronic devices.

Office hours (in 376 Altgeld Hall)
Monday 29 March, 4-5pm; Tuesday 30 March, 4-6pm; Wednesday 31 March, 11:30am-1:30pm.

Material
Sections 3.1-3.6 (see the daily schedule), plus the quiz preparation problems, and Homeworks 4,5,6,7 (including Iode Projects III).
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 xc for undamped, underdamped, critically damped or overdamped oscillations, and the response xp 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. See the Oscillator Summary handout. 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 xp(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 (for which you need to memorize the formula), we first find yc and then find yp, 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?
Note: We know how to find yc for two types of equation: constant coefficient, and equidimensional.

Memorize the solutions of the three differential equations:

• dx/dt=ax (solution x=Ceat)
• dx/dt=ax+b where a and b are constants (solution x=Ceat-b/a)
• d2x/dt2+w2x=0 where w is constant (solution x=c1cos(wt)+c2sin(wt) )

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.

Summarize what you learned in Iode Project III. The project is examinable!

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).