421:06
SOLUTIONS: ASSIGNMENT 11
SPRING 2015
13.6.1 The given problem,
kuxx ut = 0;
0 < x < 1,
t > 0;
(PDE)
u(0, t) = 100, u(1, t) = 100, t > 0;
u(x, 0) = 0, 0 < x < 1,
(BC)
(IC)
has inhomogeneous boundary conditions. We seek a particular solution of the f
MATH 421 Formula Sheet
Trigonometric Identities:
1. cos2 x + sin2 x = 1 2. tan2 x + 1 = sec2 x
1 cos 2x
1 + cos 2x
3. sin2 x =
4. cos2 x =
2
2
5. sin 2x = 2 sin x cos x 6. cos 2x = cos2 x sin2 x
7. sin(a b) = sin a cos b sin b cos a
8. cos(a b) = cos a co
Dr. Z.s Calc5 Homework assignment 7
1. Show that the given functions are orthogonal on the given interval.
2
f1 (x) = ex x5
2
f2 (x) = ex (x4 + x2 ) ,
,
[1, 1] .
2. Determine whether the following set of functions constitute an orthogonal set on the inter
Dr. Z.s Calc5 Homework assignment 2
In Problems 1-6, nd the inverse Laplace transforms.
1.
1
s4
L1
2.
L1
2
6
2
3
s
s
L1
(s + 1)2
s3
3.
4.
L1
s2
1
+ 2s 3
5.
L1
6.
L1
s3
1
+ 5s
1
(s 1)(s + 1)(s + 2)
In Problems 7-11, use the Laplace Transform to solve the g
Dr. Z.s Calc5 Homework assignment 1
1. Use the denition of the Laplace Transform to compute Lcfw_f (t) if
f (t) = t + 2
2. Use the denition of the Laplace Transform to compute Lcfw_f (t) if
f (t) = 4et + 2et
3. Use the denition of the Laplace Transform to
Dr. Z.s Calc5 Cheatsheet
FINAL VERSION
[Note: This is the only sheet allowed in any of the quizzes and exams. No calculators of course!]
Version of Dec. 12 ,2011 (adding Eulers fromula eiz = cos z + i sin z and cos z =
iz
iz
sin z = e e ).
2i
eiz +eiz
,
2
Answers to Dr. Z.s Math 421 (Calc5) Homework assignments (when applicable)
Disclaimer: not responsible for any errors. The rst nder of any error will get $1.
Version of Dec. 7 2011: (found by Tom Hansen): Changing U to Ux in one of the terms of
assignment
Second Second Chance Club II Assignment
Note: These problems are modeled after the problems in Exam II of Section 2. You are welcome
to look at the posted solution of that exam in
http:/www.math.rutgers.edu/eilberg/calc5/mt2s2Sol.pdf
z
1a Find the general
First Second Chance Club II Assignment
Note: These problems are modeled after the problems in Exam II of Section 1. You are welcome
to look at the posted solution of that exam in
http:/www.math.rutgers.edu/eilberg/calc5/mt2s1Sol.pdf
z
1a Solve (from scrat
Dr. Z.s Shortcut Methods for Solving Boundary Value Problems for PDEs
By Doron Zeilberger
Fourier Series (over (, )
Every function dened on the interval (, ) can be written as a nite or (more often innite)
linear combination of pure sine-waves and pure co
Dr. Z.s Calc5 Lecture 23 Handout: Numerical Solutions of Partial Dierential Equations
By Doron Zeilberger
Important Denitions: Discretization
The discrete approximations of the second derivatives with mesh-size h are:
uxx
1
[u(x + h, y ) 2u(x, y ) + u(x
Dr. Z.s Calc5 Lecture 22 Handout: Numerical Solutions of Ordinary Dierential Equations
By Doron Zeilberger
Important Method: (Eulers method for solving a rst-order ode)
For the initial value problem
y = f (x, y ) ,
y (x0 ) = y0
,
with mesh-size h, you den
Dr. Z.s Calc5 Lecture 21 Handout: Fourier Transform
By Doron Zeilberger
Important Denition: The Fourier Transform and The Inverse Fourier Transform
Fourier Transform:
f (x)eix dx = F () .
Fcfw_f (x) =
Inverse Fourier Transform:
F 1 cfw_F () =
1
2
F ()eix
Dr. Z.s Calc5 Lecture 20 Handout: Fourier Integrals
By Doron Zeilberger
Important Denition: Fourier Integral Representation
The Fourier Integral Representation of a function f (x) dened on the real line (, ) is
given by
1
[ A() cos x + B () sin x ] d ,
0
Dr. Z.s Calc5 Lecture 19 Handout:
Applications of the Laplace Transfrom for solving Partial Dierential Equations
By Doron Zeilberger
Version of Nov. 28, 2011 (thanks to Chris Farina, who found some typos)
Important Formula
Recall that the Laplace Transfor
Practice Exam 2, 25 April 2012, Math 421 Prof. Tumulka
1
234567
8 9 10 11 12
Name:
You may use your own formula sheet.
1) [worth 2 points out of 100] Draw the graph of f (x) = xU (1 x), where U is the
unit step function.
2) [6 points] Compute the gradient
Solutions to Dr. Z.s MATH 421 (2) Exam 2, Tue., Nov. 22, 2011, 12:00noon-1:20pm,
SEC 216
1. (15 pts.) Find the general expression, in polar coordinates, for the steady-state temperature
u(r, ) in a circular plate of radius 3, if the temperature on the cir
Solutions to Dr. Z.s Math 421(1), Exam #1
1. (15 points) Using the denition nd the Laplace transform Lcfw_f (t) (alias F (s) of
1,
if 0 t 2;
3, if t 2.
f (t) =
Sol.:
=
est 2 est
+3
s 0
s
Ans. to 1: 1
s
=
2
est (3)
est (1) +
2
0
0
2
est f (t) dt =
Lcfw_f
Solutions to Dr. Z.s Math 421(2), Exam #1
1. (15 points) Use any method to compute Lcfw_f (t) if
f (t) = (et + 1)(e2t + 1) + t3 (t + 1)
.
Sol. First use algebra to expand:
f (t) = e3t + et + e2t + 1 + t4 + t3
.
Now use the table to get
Lcfw_f (t) = Lcfw_e
640:421:06
REVIEW PROBLEMS FOR EXAM 1
SPRING 2013
Note that no books, notes, or calculators may used during the exam.
You will be given a table of the Laplace transform, based on Table III in our text.
Some unneeded formulas will be omitted, and the formu
640:421:06
ASSIGNMENT 7
SPRING 2013
Turn in the starred problems, and only the starred problems, at the beginning of the
class on Friday 3/15/2013. If your homework contains several sheets it must be stapled.
All problems are from Zill and Wright.
Section
Table of Laplace Transforms
f ( t)
1
s
1
s2
n!
,
sn+1
n a positive integer
1. 1
2. t
3. t
f ( t)
Lcfw_f (t) = F (s)
n
4. sin kt
5. cos kt
6. sin2 kt
7. cos2 kt
8. eat
9. sinh kt
12. eat tn
Lcfw_f (t) = F (s)
n!
,
(s a)n+1
n a positive integer
13. eat sin
640:421:06
REVIEW PROBLEMS FOR EXAM 1
SPRING 2013
Note that no books, notes, or calculators may used during the exam.
You will be given a table of the Laplace transform, based on Table III in our text.
Some unneeded formulas will be omitted, and the formu
640:421:06
ASSIGNMENT 4
SPRING 2013
Turn in the starred problems, and only the starred problems, at the beginning of the
class on Friday 2/22/2013. If your homework contains several sheets it must be stapled.
All problems are from Zill and Wright, and the
ASSIGNMENT 1 SPRING 2013
Turn in the starred problems, and only the starred problems, at the beginning of the
class on Friday 02/01/2013. If your homework contains several sheets it must be
stapled. All problems are from Zill and Wright.
Section 4.1: 1, 3
ASSIGNMENT 3 SPRING 2013
Turn in the starred problems, and only the starred problems, at the beginning of the class
on Friday 02/15/2013. If your homework contains several sheets it must be stapled. Note
the additional problems 3.A and 3.B, which are to b
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v ty p
d
r
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r y g d v ur p PW o
r y
dt
vy
r d cfw_ur 9rz
y
r
dt p
| d cfw_ur rzXDedc`
v ty p f b
d
v d ty f b
d cfw_ur p PW o Dedc`
vy p o
d t sqSW
v t y sp SW o
d
vy p
d t srz
x
y l
y l e
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eTFsDsDRE
Q F~ H A~ E F
Practice Exam 2, 3 December 2012, Math 421 Prof. Tumulka
1
234567
8 9 10 11 12
Name:
You may use your own formula sheet.
1) [worth 4 points out of 100] Is f odd, even, or neither? (No justication required.)
a) f (x) = sin x cos x
b) f (x) = sin x + cos x