HOMEWORK 1 SOLUTIONS
Section 1.1 : #2, 3, 11, 12
#2) Which of the following operators are linear? (This is accomplished by
checking if Lu = cLu and L(u + v) = Lu + LV are satised).
(a) Lu = ux + xuy
(b) Lu = ux + uuy
(c) Lu = ux + u2
y
(d) Lu = ux + uy +

Math 4570-01
Test 1 (40 points possible)
September 21, 2015 (Monday)
Name:
Please justify your work in order to receive full credit.
1. (5 points) Show that the inhomogeneous rst order dierential equation
du
dx
+ p(x)u = q(x) is linear.
Solution: An inhom

A First Course in Quasi-Linear Partial
Dierential Equations
for Physical Sciences and Engineering
Solution Manual
Marcel B. Finan
Arkansas Tech University
c All Rights Reserved
2
Preface
This manuscript provides the complete and detailed solutions to A Fi

Math 4570-01
Review Problems for Test 2
Wednesday, October 28
Name:
Please show all work relevant to the solution.
1. A wave f (x + ct) travels along the semi-innte string (x > 0) for t < 0 subject to the boundary
condition u(0, t) = 0 for all t. Find the

Math 4570-01
Review Problems for Test 1
Friday, September 18
Name:
Please show all work relevant to the solution.
1. Practice with linearity.
a. Is f (x, y) = 2xy + 7y + y sin x linear in x, linear in y, linear in x and y, or none of these? Justify
your a

ECS 332: Principles of Communications
2012/1
HW 1 Due: July 13
Lecturer: Prapun Suksompong, Ph.D.
Instructions
(a) ONE part of a question will be graded (5 pt). Of course, you do not know which part
will be selected; so you should work on all of them.
(b)

Math 462: HW3 Solutions
Due on August 1, 2014
Jacky Chong
1
Jacky Chong
Remark: We are working in the context of Riemann Integrals.
Problem 1
2.4.1 Solve the diusion equation with the initial condition
(x) = 1
|x| < l
for
and
(x) = 0
|x| > l.
for
Write yo

2
Linear Equations: Solutions and
Approximations
In the last chapter we studied autonomous rst-order DE models and a few
elementary techniques to help understand the qualitative behavior of these
models. At this point, the reader should be able to solve t

HOMEWORK 5 SOLUTIONS
Section 3.3
1
Well solve the inhomogeneous diusion equation on the half-line with Dirichlet boundary conditions using the reection method. The setup of the problem is as follows:
u kuxx = f (x, t) 0 < x < , 0 < t <
u(0, t) = 0
u(x, 0

Matlab Starter Kit
Math 355 - Dierential Equations, Spring 2004
by Brody Johnson
What follows is a very basic guide designed to help acquaint new users with Matlab. Matlab
provides an extremely useful computational platform for mathematics and engineering

Math 4570-01
Homework 7
Due October 28 (Wednesday)
Name:
Please show all work relevant to the solution.
1. A full Fourier series for (x) on x
has the form
(x) = a0 +
an cos
n
x + bn sin
n
x
.
n=1
This expansion uses the pairwise orthogonality of the fami

Math 4570-01
Homework 6
Due October 16 (Friday)
Please show all work relevant to the solution.
Name:
1. (5 points) Use separation of variables to nd a series solution of
ut = kuxx
subject to
u(0, t) = 0,
ux ( , t) = 0,
& u(x, 0) = (x)
over the domain 0 <

Math 4570-01
Homework 7
Due October 28 (Wednesday)
Name:
Please show all work relevant to the solution.
1. A full Fourier series for (x) on x
has the form
(x) = a0 +
an cos
n
n
x + bn sin
x
.
n=1
This expansion uses the pairwise orthogonality of the fami

Math 4570-01
Homework 5
Due October 12 (Monday)
Name:
Please show all work relevant to the solution.
1. A wave f (x + ct) travels along the semi-innte string (x > 0) for t < 0. Find the vibrations of the
string for t > 0 if ux (0, t) = 0 for all t. Simpli

Math 4570-01
Homework 5
Due October 12 (Monday)
Please show all work relevant to the solution.
Name:
1. A wave f (x + ct) travels along the semi-innte string (x > 0) for t < 0. Find the vibrations of the
string for t > 0 if ux (0, t) = 0 for all t. Simpli

Math 4570-01
Homework 4
Due October 2 (Friday)
Name:
Please show all work relevant to the solution.
1. Show that the wave equation has the following invariance properties.
a. Any translate u(x y, t), where y is xed, is also a solution.
b. Any derivative,

Math 4570-01
Homework 4
Due October 2 (Friday)
Name:
Please show all work relevant to the solution.
1. Show that the wave equation has the following invariance properties.
a. Any translate u(x y, t), where y is xed, is also a solution.
Solution: Let v(x,

Math 4570-01
Homework 3
Due September 18 (Friday)
Name:
Please show all work relevant to the solution.
1. Consider the one-dimensional wave equation, utt = c2 uxx , subject to
u(x, 0) = (x),
ut (x, 0) = (x),
x R.
(a) Prove that this initial-value problem

Math 4570-01
Homework 3
Due September 18 (Friday)
Name:
Please show all work relevant to the solution.
1. Consider the one-dimensional wave equation, utt = c2 uxx , subject to
u(x, 0) = (x),
ut (x, 0) = (x),
x R.
(a) Prove that this initial-value problem

Math 4570-01
Homework 2
Due September 11 (Friday)
Name:
Please show all work relevant to the solution.
1. Solve the rst-order equation ut 3ux = 0 with the auxiliary condition u(0, x) = sin x when t = 0.
Solution: Interpreting the PDE as a directional deri

Math 4570-01
Homework 1
Due September 2 (Wednesday)
Please show all work relevant to the solution.
Name:
1. Consider the inhomogeneous rst-order dierential equation
du
2xu = x.
dx
Write the equation in the form Lu = f (x) where L is an operator. Is L lin

Math 4570-01
Homework 1
Due September 2 (Wednesday)
Name:
Please show all work relevant to the solution.
1. Consider the inhomogeneous rst-order dierential equation
du
2xu = x.
dx
Write the equation in the form Lu = f (x) where L is an operator. Is L lin

Chapter 3
Heat equation
3.1
Averaging
Here are a couple of calculations related to the heat equation. Weve found the
solution on the line to be
u(t, x) =
1
4kt
exp(x y)2 /4kt)(y) dy.
We want to show that the solution of the heat equation is obtained by a

Math 4570-01
Extra Credit 1 (10 homework points)
Due by November 2nd, 2015 (Monday)
Please show all work relevant to the solution.
Name:
The convolution operation played a central role in our solution of the homogeneous diusion equation on
the line and wa

Math 4570-01
Bonus Problems for Test 1
Due September 30 (Wednesday)
Name:
Please show all work relevant to the solution.
1. (2 points) Consider a traveling wave u(x, t) = f (x at) where f is a given function of one variable.
(a) If u is a solution of the

Math 4570-01
Bonus Problems for Test 1
Due September 30 (Wednesday)
Name:
Please show all work relevant to the solution.
1. (2 points) Consider a traveling wave u(x, t) = f (x at) where f is a given function of one variable.
(a) If u is a solution of the