Bad copies of Exam 3 A-D and E-K were posted on the internet Nov 1,
discovered Nov 6 and replaced. If you got the exam in class (colored
paper), then you have the right exam. Otherwise, replace it. Typos will
be posted. So far, two annoying ones have been found, version E-K and
L-R, problem 2(b) and A-D and L-R problem 3. Notes below were obtained
by conversations and office visits.

Problem 1.

Use (33) page 335.
No answer checks required for problem 1. Not much maple either.

1(a): See page 279 for the definition of Wronskian.

1(b): Use y=yh+yp superposition. Solve the homogeneous eq. Find the
basis elements. Remove all terms from the given y that are constants
times these basis elements. If a trig identity is required, apply it.

1(c): Formula (33) can be written with definite integrals instead
of indefinite integrals. Choose lower limit x0 and upper limit x.
Likely, you will choose x0 = 1, for example, on the E-K exam3.

1(d): Replace the RHS of the DE by 1. Solve the problem for this
simplified case. Then deal with the yp part for the problem's case by
using the variation of parameters formula. The question reduces then to
finiteness of some integral.


Problem 2.

2a. Report f = f1 + f2 + f3 + ... where each f1, f2, f3 is an atom.
Identify atoms and types in a table. For example, f = x + sin(x)
decomposes to f = f1 + f2 where

                          Atom            Type
                          f1=x            (1)
                          f2=sin(x)       (4)

2b. Please correct these typos:
     Ver L-R: remove extra term d5 sin(3x) in 2(b).
     Ver E-K: remove extra term d6 sin(3x) in 2(b).
Assumed here is that y1, y2, y3 are predicted trial solutions before the
fixup rule has been applied, that is, without the x^s factor present in
the book's table page 331. The corrected trial solution is to be
reported, namely, add the missing x^s factor. Use the web notes on the
Table and Kummer methods (4 pages, next to exam pdf sources).
    
2c. Use the table page 331 or the web notes on the table and Kummer's 
method to find a fully corrected trial solution. Do not solve for the 
undetermined coefficients!

2d. Stuff the given trial solution into the given DE and solve for the
undetermined coefficients. Yes, you should check your answer, it has to
work in the DE.


Problem 3. To recover from Green L-R and Yellow A-D errors,
please change the DE in #3 to x'' + 4x' + 68x = 12 cos(wt),
the same as the pink exam. The Pink and Blue exams need no
changes.

3a. Use (19) page 346.

3b. Use formula (21) page 346. Make a graphic like Fig 5.6.9 page 348.

3c. Use w* = (1/m) sqrt(km - c^2/2), which is the root of C'(w)=0
    when practical resonance occurs.

3d. Let w = w* + 0.1. 
    (3d.1) Graph the answer from (3a).
    (3d.2) Solve for x(t)=xh(t)+xp(t) using maple dsolve().
           Compare the maple answer to the answer in (3d.1).
           Check the roots of the char eq against maple's x(t).

Problem 4.

4a. This college algebra example applies:

        Given a quadratic has roots r=1, r=2, find the quadratic.
        Solution: Both r-1 and r-2 are factors by the factor theorem,
        therefore (r-1)(r-2)=0 is the quadratic.

The characteristic equation r^2+(R/L)r + 1/(LC)=0 is a quadratic.
It can be found explicitly from the roots of the characteristic
equation, which appear in the recipe solution Qh(t), by following
the preceding example. This gives equations for R/L and 1/(LC),
from which all the missing constants are found.

To find E, stuff Qp(t) into the DE.

4b. Formulas from the textbook apply, namely (4)-(10) pp 340-341.
Refer to them in solving for Q(t), which is obtained directly from
(9) page 341. The A in the problem is a constant whereas the symbol
A(t) appears in the text - do not confuse the two symbols.