M412 Exam 1 Practice Problems (not to be turned in)
In addition to these problems, you should of course consider all previously assigned homework problems to
be good practice.
Separation of variables
1. Establish the following integral identities:
L
0, n
M412 Assignment 9 Solutions
1. [10 pts] Use separation of variables to show that solutions to the quarter-plane problem
t > 0, 0 < x <
ut = uxx ;
u(t, 0) = 0
|u(t, +)| bounded
u(0, x) = f (x), 0 < x < ,
can be written in the form
u(t, x) =
0
2
C ( )e t s
M412 Assignment 8 Solutions, due Friday November 11
1. [10 points] Finish our proof of Fouriers Theorem by showing that
1
N 2L
N
x +L
f (y ) 1 + 2
lim
x
cos(
n=1
1
n
(y x) dy = f (x+ ).
L
2
Solution. Proceeding similarly as we did in class, set z = y x to
M412 Assignment 7 Solutions, due Friday November 4
1. [15 pts] (Mean Value Property in three space dimensions.) Suppose is an open subset of R3 and
u C 2 () solves the Laplace equation in . Show that if (x0 , y0 , z0 ) , and Sr (x0 , y0 , z0 ) is a sphere
M412 Assignment 6 Solutions, due Friday October 28
1. [10 pts] Show that for the eigenvalue problem
a x b,
(p(x)ux )x + q (x)u + (x)u = 0;
eigenvalues are related to their eigenfunctions u by the Rayleigh quotient
=
b
(p(x)u2
x
a
q (x)u2 )dx + (p(a)u(a)u
M412 Assignment 5 Solutions
1. [10 pts] Haberman 2.5.1 (a).
Solution. Setting u(x, y ) = X (x)Y (y ), we nd that X and Y satisfy
X + X = 0
Y Y = 0,
with boundary conditions
X (0) = 0
X (L) = 0
Y (0) = 0.
In this case, we nd from the X equation that the ei
M412 Assignment 4 Solutions
Two errors in the Practice Problems for Exam 2 have been brought to my attention. In Problem 3, f (x)
should be given explicitly as x2 . Also, in the solution to Problem 3, the value of should be 2.
1. [10 pts, 5 pts each] Habe
M412 Assignment 3 Solutions
1. [10 pts] Use the method of characteristics to solve the PDE
ux uy + 2y =0
u(x, y ) = xy on the line x + 2y = 1.
Solution. In this case, set U (t) = u(x(t), y (t) and choose
dx
= 1; x(0) = x0 x = t + x0
dt
dy
= 1; y (0) = y0
M412 Assignment 2, due Friday September 9
1. [10 pts] Use the method of diagonalization to determine a general solution for the ODE system
3
y1 = y1 + y 2
4
y2 = 5 y1 + 3 y2 .
Solution. In matrix form, this equation has the form
y1
y2
3
4
1
5
=
or y = Ay
Assignment 1, Solutions
1. [5 pts] We solve this problem by separation of variables. We have
dy
= 3x2 dx
+1
y2
dy
=
+1
3x2 dx tan1 y = x3 + C y (x) = tan(x3 + C ).
y2
Using the initial condition y (0) = 1, we have 1 = tan(C ), so that C = k , k = 1, 2, .
M412 Practice Problems for Exam 3
Qualititative properties of the Laplace equation
1. For the Laplace equation
0 x 1,
uxx + uyy = 0;
u(x, 0) = 10x(1 x);
u(0, y ) = 10y (1 y );
0 y 1,
u(x, 1) = 1 x
u(1, y ) = 1 y,
nd upper and lower bounds on u(x, y ) in t
M412 Practice Problems for Exam 2
Please keep in mind that Exam 2 will be given Wednesday, October 19 7:009:00.
Equilibrium Problems
1. Haberman 1.4.1 Parts (d) and (h).
2. Haberman Problem 1.4.11.
3. For the partial dierential equation
ut = uxx + 4x
ux
M412 Practice Problems for Final Exam
1. Solve the PDE
u t + t3 u x = u
u(t, 0) = t,
t>0
u(0, x) = 1 ex ,
x > 0.
2. Solve the PDE
utt = c2 uxx ; x > 0, t > 0
u(0, x) = f (x); x > 0
ut (0, x) = g (x);
ux (t, 0) = t;
x>0
t > 0.
3. Solve the PDE
uxx + uyy =