Mathematics 421 Essay 2 An operational view of Fourier coefcients Spring 2008
0. Introduction Although Laplace transforms were dened using an integral containing a parameter, the computation of transforms and inverse transforms used a special list of prop

Mathematics 421 Solution to ch. 4 rev 36 Spring 2008
Introduction
Many students have difculty solving differential equations with data dened by cases. Before attempting such a problem, it is useful to consider how to interpret the question in order to kno

Mathematics 421 Essay 4 Partial Differential Equations Spring 2008
0. Introduction
The Method of Separation of Variables is used to construct special solutions of some linear partial differential equations. Other solutions can then be found as linear comb

Mathematics 421 More Solutions Chapters 12 and 13 Spring 2008
Introduction
To aid preparation for the second exam, some complete solutions of problems have been prepared. The solutions of 12.2#9 and 12.3#22, as well as the solutions of the problems in the

Math 421:01 and 02 Spring 2008 Final Exam Review and Formula Sheet
Trigonometry The formula
e i t D cos t C i sin t
(with i 2 D 1) connects exponential and trigonometric functions. Consequences include the basic identities cos. C / D cos cos sin sin
sin.

Math 421:01 and 02 Spring 2008 Second Review and Formula Sheet
Trigonometry The pair .cos t; sin t / gives the parametrization of the unit circle by arc length starting from the positive x axis. Thus, these functions are periodic with period 2 and some sp

Math 421:01 and 02 Spring 2008 Formula Sheet
Trigonometry The formula
e i t D cos t C i sin t
(with i 2 D 1) connects exponential and trigonometric functions. Consequences include the basic identities cos. C / D cos cos sin sin sin. C / D sin cos C cos si

Mathematics 421 Extra Fourier Series Solutions Spring 2008
Introduction Few solutions caught the intent of the exercises in the supplement entitled, An operational view of Fourier coefcients, so solutions will be given here. The instructions specied that

Mathematics 421 Solutions 12.2 and 12.3 Spring 2008
Introduction
A student asked about Problem 9 in Section 12.2 in ofce hours. The solution has some unusual features, so it is useful to share it with the whole class. Problem 22 of Section 12.3 gave more

Mathematics 421 Solutions 13.6 Spring 2008
The use of separation of variables to identify solutions of a partial differential equation that are products of functions of the individual variables relies on the equation being linear and homogeneous to constr

Mathematics 421 Solutions 13.4, Number 6 Spring 2008
The problem
Solve uxx D u t t by dAlemberts method with the initial conditions
u.x; 0/ D 1 ; 1 C x2 u t .x; 0/ D 0:
and graph the result for t D 0; 1; 3.
Analysis
Then
We write
u.x; t / D F .x C t / C G

Math 421:02 Final Exam Prof. Bumby May 14, 2008 Noon 3 PM
This is a blue book exam. Write your name on the cover of your blue book. The other requested information is not needed, and should not be given. All work should be in the blue book, organized so t

Math 421:01 Final Exam Prof. Bumby May 13, 2008 Noon 3 PM
This is a blue book exam. Write your name on the cover of your blue book. The other requested information is not needed, and should not be given. All work should be in the blue book, organized so t

Math 421:01&02 Prof. Bumby Exam 2 April 09, 2008 Name
All class exams will be graded on the basis of 100 points. On this exam, those points are distributed over 6 problems (on 5 pages). No books or papers other than the ofcial formula sheet may be used, b

Math 421:01&02 Prof. Bumby Exam 1 February 20, 2008 Name
All class exams will be graded on the basis of 100 points. On this exam, those points are distributed over 7 problems (on 7 pages). No books or papers other than the ofcial formula sheet may be used

Mathematics 421 Essay 3 Boundary Value Problems Spring 2008
0. Introduction A second order differential equation has a general solution containing two parameters. Typically these parameters are the values of the solution and its rst derivative at a single

Mathematics 421 Essay 1 Using the Laplace transform Spring 2008
0. Introduction
dened by
The Laplace transform of a function of t is a function of a new variable s
Z ff .t /g D
0 1
f .t /e
st
dt
This is an improper integral, so convergence must be conside

Mathematics 421 Solutions 4.4 Spring 2008
Introduction
Exercises 10 and 14 involved many steps in a single problem, so there were few solutions that were completely correct. These solutions are presented as an example of how the work on such problems may