Tuesday 3 April, in class, worth 15%.
You may not use books, notes, or electronic devices on the test.
Study session in 141 Altgeld: Monday 2 April, 5-6pm
Material
Sections 3.4-3.6, 4.1 and 5.1-5.5.
What and how to study
First make summary notes of the important ideas and methods
from each section.
You should aim to feel confident solving any
"reasonable" nonhomogeneous problem, using the method of undetermined
coefficients or the method of variation of parameters (for which you should
learn the formula for yp).
In both undetermined coefficients and variation of parameters,
you first find yc and
then find yp, then apply the initial conditions to y=
yc+yp, in order to evaluate the
constants in yc. Note: We know how to find yc for two
types of linear equation: constant coefficient, and equidimensional.
Ignore Sec. 3.5 Rule 1 as it is stated in the textbook.
It is much better to use Rule 1 as stated in the
"Undetermined Coefficients Summary" on the
handouts page.
Summarize the main properties and results
in Sections 3.4 and 3.6, on mechanical vibrations (there are four cases, since
you can either have damping or not, and forcing or not; note we always assume
the forcing function is periodic or a linear combination of periodic terms).
Recall the two handouts from class that summarize aspects of these sections.
Work through the derivations of the solution formulas, in the four cases.
Write short explanations of...what is beating? resonance? practical resonance?
In what situations do they occur? (e.g. beating occurs in the undamped,
forced case, and is due to interactions between xc and
xp, both of which are periodic and do not decay, in this case.)
Re-work all homework problems, and quiz problems. Attempt the Practice Test
(on main webpage).
Note. Problem 7 on the Practice Test involves material we have not
covered yet (we will cover it later).
Also, the Practice Test provides no practice on Chapters 4,5.
Understand the meaning of the theorems and definitions, because they give structure to the examples that we consider. I will not ask you to prove any theorems.
If you just work problems without first making summary notes, you will probably be wasting your time, because you won't have a mental framework into which to fit the examples you are working. It is the underlying methods and techniques that we care about; we we work examples in order pwd to check that we understand the techniques.