HOMEWORK 3 SOLUTIONS
Section 2.1 : #2, 7
#2) Solve
utt = c2 uxx
u(x, 0) = log(1 + x2 )
ut (x, 0) = 4 + x
Answer: Using the solution to the initial value problem wave equation, we
have
u(x, t) =
1
1
l
MATH 110 MIDTERM WINTER 2010
Please justify all your steps!
1. (a) Find the general solution of the transport equation ux + 2xey uy = 0.
(b) Find the particular solution of (a) satisfying u(x, 0) = x2
Math 110A Midterm Solutions
C.T. Wildman
1. (a) The characteristic curves satisfy ODE
dy
= 2xey
dx
Separating variables and integrating, we obtain
Z
Z
y
e dy = 2x dx
Then ey = x2 +C or C = x2 +ey . Th
Setion 2.3, Problem 4
For this problem it is neessary to use the strong maximum priniple whih is mentioned
p. 41. This simply says that if u is not onstant, then one an replae 6 in the maximum
prinipl
Here are more homework solutions. If you want to know more about other homeworks, or
you would like to see more details or have other questions, ask in lass, review setion or oe
hour.
Setion 3.1, Prob
Math 110A Homework Solutions
C.T. Wildman
6.3.3 Solve uxx + uyy = 0 in the disk cfw_r < a with the boundary condition
u = sin3
on r = a
Solution
We use the identity sin(3) = 3 sin 4 sin3 to write
u(a
1. Setion 1.1, Problem 3
Reall that an equation is linear homogeneous if you an add two solutions and get another.
A good way to hek for this is simply to see wether or not the equation involves any m
Math 110A Practice Final Solutions
C.T. Wildman
1. The characteristic curves satisfy the ODE
dy
= yex
dx
Separating variables, we obtain y 1 dy = ex dx, so an integration yields
log y = ex + C
Notice
MATH 110 MIDTERM SOLUTIONS
1. We obtain the characteristic curves by solving the differential equation
y0 =
y
.
x
R 1
R
Separating variables, we obtain y1 dy = x dx, from which we deduce log y =
log x
Setion 5.1, Problem 9
For this problem, we use the Neumann BC formula for solutions to the wave equation,
whih is similar to the Dirihlet BC formula derived in the previous HW. However, there is a
ver
Math 110A Homework Solutions
C.T. Wildman
4.3.4 Suppose a0 , al < 0 and (a0 +al ) < a0 al l. Show that there are two negative eigenvalues
for the problem X 00 = X.
Solution:
0 +al )
Following the hint
HOMEWORK 10
6.2.1. Following the hint, guess that
u(x, y ) = Ax2 + By 2 + Cxy + Dx + Ey + F.
Then,
(1) u = 0 A = B.
(2) ux (0, y ) = a CY + D = a.
(3)
ux (a, y ) = 0 2Aa + CY + D = 0.
(4)
uy (x, 0) =
HOMEWORK 11
7.2.2. Apply G2 with u = , v = 1/(4 |x|), over the domain D = B \ B .
(B = unit ball centered at zero, B = ball of radius centered at zero.)
Then v = 0 in D and we get
1
dx =
4 |x|
D
(
B
(
HOMEWORK 4 SOLUTIONS
Section 2.4 : #2, 4, 9, 11, 15
#2) Solve the diusion equation with the given initial condition:
ut = kuxx in cfw_ < x < , 0 < t <
u(x, 0) = (x) =
1
3
if x > 0,
if x < 0.
Solutio
HOMEWORK 5 SOLUTIONS
Section 3.1 : #1, 3, 4
#1) Solve the diusion equation on the half-line with the Dirichlet boundary condition:
ut = kuxx on cfw_0 < x < , 0 < t <
u(x, 0) = ex
u(0, t) = 0
Solutio
HOMEWORK 6 SOLUTIONS
Section 3.2 : #8, 10, 11
#8) For the wave equation in the nite interval (0, ) with Dirichlet conditions, explain the solution formula within each diamond-shaped region.
Solution:
HOMEWORK ASSIGNMENT
All the assignments are from the textbook, unless otherwise specied. They are
due in class, at the beginning of the lecture. No assignment is due on the rst week
of classes and on
Midterm I: Partial Dierential Equations
Name:
Problem 1 2 3 4 5 Total:
Max
15 20 20 25 20 100
Scores
TOTAL:
/100
Instructions
Time for test: 75 minutes.
Write clearly in pen or pencil, and make your
PDEs: Midterm 2
1
Midterm 2
Name:
Problem 1 2 3 4 Total:
Max
25 25 25 25
100
Scores
Instructions
Time for test: 75 minutes.
Write clearly in pen or pencil, and make your nal answer easy to nd.
To r
HOMEWORK 9
6.1.4. We look for a solution u = u(r). Thus u must satisfy
2
c1
urr + ur = 0 (r2 ur )r = 0 r2 ur = c1 u = + c2 .
r
r
From the boundary conditions
c1
+ c2 = A
a
c1
u(b) = B + c2 = B.
b
Solv