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 .
(c) Extra Credit: Find the general solution of ux +
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 . Thus, the general solution to our equation
is u(x, y) = f
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
priniple by < in the interior. That is, if u is not onstant th
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, Problem 4
For (a) and (b), we use equation (6) in Setion 3.
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, ) = sin3 =
3
1
sin sin(3)
4
4
The solution to our pro
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 multipliations or non-linear funtions of the solution (o
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 that |y| = y since y 1. Thus the general solution is
u(
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 + c. Exponentiating and setting c = ec, we see that th
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
very important dierene that has to do with the presene of
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, we define y() = (a
. Solving the equation y 0 () = 0
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) = b Cx + E = b.
(5) uy (x, b) = 0 2Bb + Cx + E = 0.
From
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
(
D
v
v )dS +
n
n
v
v )dS =
n
n
(
B
v
v )dS.
n
n
The
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.
Solution: This problem is the same as the problem at the begin
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
Solution: Using the method of odd extensions, consider the diu
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: The equation and boundary conditions we are considerin
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 the weeks of the midterms.
Week 2.
Written assignment:
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 nal answer easy
to nd.
To receive any partial credit
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 receive any partial credit you must clearly show your wo
PARTIAL DIFFERENTIAL EQUATIONS, MATH V3028
Course Webpage. http:/www.math.columbia.edu/~ desilva/pde/
Instructor. Daniela De Silva
Email: desilva@math.columbia.edu
Oce: Rm 518, Columbia Math Department
Oce hours: Tu 4,30pm-5,30pm, Wed 4pm-5pm (or by appoi
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
Solving for c1 , c2 we obtain
u(a) = A
c1 = (B A)(1/a 1/b)