Engineering Math
Homework 8
Fall 2012
(due Thursday, November 15)
Consider the function 0 , which is defined as follows on the interval 1 1 and then made
#
periodic with period 1.
0 B cosh B
1 B 1
#
#
11
(1) Graph this periodic function 0 at least on the

Engineering Math
Homework 7
Fall 2012
(due Thursday, November 8)
Consider the following differential equation.
" B# Cww BCw :# C !
B! !
Note that ! is an ordinary point of this differential equation, so there are power series solutions
about !. Note also

Engineering Math
Homework 6
Fall 2012
(due Thursday, October 25)
"#
p
Let F ! C# / % C /B , and let W be the surface given as follows:
B# D # " C#
%
B!
!C'
D!
oriented upward
p
Find the value of the surface integral ' ' F p .W . Show complete, clear, and

Engineering Math
Homework 5
Fall 2012
(due Thursday, October 18)
p
p
Let F be the vector function in the plane given by F B C C C% . Let G be the closed
curve which first follows the straight line segment from " ! to ! " and then follows the
#
p
"
curve C

Engineering Math
Homework 4
Fall 2012
(due Thursday, October 4)
(1) Consider the vector function p > sinh > > cosh >. Find the length of the arc
r
determined by this vector function from the point where > ! to the point where > ".
Hint. Simplify before in

Engineering Math
Homework 3
Fall 2012
(due Thursday, September 27)
$
!
Consider the matrix A !
!
!
'
!
!
!
!
* "# "&
'
'
"#
$
$
'
.
!
!
'
!
!
$
There are several vector spaces associated with A, namely, its row space, its column space, its
null space, a

Engineering Math
Homework 2
Fall 2012
(due Thursday, September 13)
Use Laplace transforms to solve the following integrodifferential equation for the unknown
function C>.
Cw > "'! C:.: ># )>
>
C! !
Hint 1. After you have found ] =, it would be wise to che

Engineering Math
Homework 1
Fall 2012
(due Thursday, September 6)
(1)
Let <>
!
"$/$>
>"
.
>"
Find _<>.
Hint. Before beginning the second problem, it would be wise to check with a friend or with me
to be sure your answer to the first problem is correct.

Engineering Math
Homework 9
Fall 2012
(due Monday, November 26)
Refer back to Homework 7. You may wish to look at your own homework paper (which has
now been returned to your Pendaflex folder), and you may also wish to refer to the Solutions
which have be

Engineering Math
Homework 10
Fall 2012
(due Thursday, December 6)
Consider the following initial boundary value problem.
#
)! ` ? `?
`B#
`>
?! > !
for ! B % and > !
and
?B ! 0 B
?% > !
for ! B %
for > !
where 0 B is specified below
(1) Write out the form

EnMath B, ESE 319-01, Spring 2015
Lecture 4: Frobenius Solutions, Indicial Roots
Jan. 21, 2014
Frobenius Method (when 0 is a regular singular point). In this course we look only
for series solutions centered at 0.
y = cn x n+r
y = c n (n + r )x n + r 1
n

EnMath B, ESE 319-01, Spring 2015
Lecture 2: Power Series, continued
Jan. 14, 2015
Finding the solutions. Once we determine that 0 is an ordinary point of the ODE, we
assume a power series solution for y centered at 0 and take its first two derivatives.
y

EnMath B, ESE 319-01, Spring 2015
Lecture 1: Power Series Intro. (Zill Chapter 5)
Jan. 12, 2015
Definition of a Power Series. This is a power series.
n
2
y (x ) = c n (x a ) = c0 + c1 (x a ) + c 2 (x a ) + !
n =0
This power series is said to be centered a

EnMath B, ESE 319-01, Spring 2015
Lecture 7: Bessel Functions
Jan. 28, 2015
The Bessel Equation. The Bessel equation of order , where > 0.
(
)
x 2 y + xy + x 2 2 y = 0
From before:
Frobenius series (0 is a regular singular point).
IC = (0, ).
Indicial

EnMath B, ESE 319-01, Spring 2015
Lecture 8: Legendre Functions, Orthogonal Functions Intro.
Feb. 2, 2015
The Legendre Equation. The following equation, the Legendre equation, comes up in
various physics and engineering applications, especially boundary v

Review of Partial Fractions
The purpose of the method of partial fractions is to express a fraction having a complicated
polynomial denominator as the sum of fractions whose denominators are simpler polynomials.
Step 0. Make sure that the degree of the nu

Review of Basic Matrix Definitions and Properties
A matrix is a rectangular array of numbers. A matrix A with 7 rows and 8 columns is said to
have size 7 8. The entry in the 3th row and the 4th column of A is denoted +34 , and the matrix
A can be denoted

Integration Formula List
Let + ! and , !.
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
' B cos +B .B B sin +B
+
"
+# cos +B
' B sin +B .B B cos +B
' B# cos +B .B
+
"
+# sin +B
B#
#B
#
+ sin +B +# cos +B +$ sin +B
' B# sin +B .B B# cos +B
' /+B cos ,B .B
' /+B sin

Formulas and Values You Should Know
Integration (inside front cover of textbook)
#14, 68, 1516
Exponents and Logarithms
/B
/C
/B /C /BC
/BC
/B C /BC
ln B ln C ln B
C
ln B ln C lnBC
lnB< < ln B
ln " !, ln / "
Trigonometry
cos# B sin# B "
cosB cos B
sinB