Math 4470/5470 Homework #2 Solutions
1. Using the techniques discussed in class, show how to reduce the PDE
4uxx + 12uxy + 10uyy 4ux + 2uy = 0
to the form
v + v + Cv = 0
for some constant C . What is C in this case?
Solution: We rst think of it in terms o
Math 4470/5470 Homework #6 Solutions
1. Find the Green function for the problem u (x) + u(x) = 0 with boundary conditions
u (0) = 0 and u (1) = 0. In other words solve the problem
u (x) + u(x) = 0
for 0 < x < 1 with x = y,
u (0) = 0,
u (1) = 0
with limits
Math 4470/5470 Homework #5 Solutions
1. It is not quite true that every eigenvalue problem f (x) = f (x) with homogeneous
boundary conditions has only negative eigenvalues: Robin boundary conditions can
break this.
(a) Show that there is exactly one posit
Math 4470/5470 Homework #10 Solutions
1. Prove the Dini formula
N
1
sin (N + 2 )
1
+
cos n =
.
1
2 n=1
2 sin 2
Hint: using Eulers formula ei = cos + i sin , write cos n = 1 (ein + ein ). Then
2
use the geometric series formula
N
rn =
n=1
rN +1 r
r1
with
Math 4470/5470 Homework #11 Solutions
1. Use the Maximum Principle for the heat equation (from Homework #10) to prove
uniqueness for the problem ut = uxx in the domain 0 x L and t > 0, with
given initial condition u(x, 0) = f (x) and given boundary condit
Math 4470/5470 Homework #12 Solutions
1. If f (x) =
the formula
n=0
an cos nx for f (x) = sin 2x on 0 x , nd the coecients an using
cos A sin B =
1
2
sin (A + B) sin (A B) .
Does the Fourier series converge? How do you know?
Solution: The formula says
a0
Math 4470/5470 Homework #8 Solutions
1. Suppose u(x, y) is a function satisfying uxx + uyy = 0 (such a function is called harmonic). A function v(x, y) satisfying the Cauchy-Riemann equations
u
v
=
x
y
u
v
=
y
x
and
is called a harmonic conjugate of u.
(a
Math 4470/5470 Homework #7 Solutions
1. Suppose we want to minimize
1
e2x v (x)2 v(x)2 dx
E=
0
over all functions v satisfying v(0) = 0 and v(1) = 1.
(a) Derive the dierential equation that the minimizing function u(x) will satisfy.
Solution: The usual te
Math 4470/5470 Homework #3 Solutions
1. Suppose that f (t) is a continuous function on R and that a is some constant. Prove
that any solution of u (t) = f (t)u(t) with u(0) = a is unique, by showing that for any
two solutions u1 (t) and u2 (t), the quotie
Math 4470/5470 Homework #2 Solutions
1. Using the techniques discussed in class, show how to reduce the PDE
4uxx + 12uxy + 10uyy 4ux + 2uy = 0
to the form
v + v + Cv = 0
for some constant C. What is C in this case?
Solution: We rst think of it in terms of
Math 4470/5470 Homework #4 Solutions
1. (A linear algebra problem) Suppose A is a symmetric n n matrix; in other words
A = A. Suppose u and v are eigenvectors of A with Au = u and Av = v where
= . Show that u v = u, v = 0. (Hint: rst show that (Au) v = u
Math 4470/5470 Homework #1 Solutions
1. Section 1.0, page 4. Exercises 12. (Specify the most general solution satisfying the
conditions; is it unique?)
Exercise 1: Integrate/solve.
(a) u (x) = 0,
< x <
Solution: u(x) = Ax + B for some constants A and B.
Math 4470/5470 Exam #1 Solutions
1. (a) (15 points) What are the three conditions that make a linear partial dierential
equation with given initial and/or boundary conditions well-posed ?
Give a one-sentence denition for each of the terms.
Solution:
Exis
Math 4470/5470 Exam #2 Solutions
1. (a) (15 points) What must a and b be for
u (x) =
a 0<x<
b otherwise
to be an approximate delta-function? Explain briey and draw a picture.
Solution: We need three conditions: u (x) 0 for all x, lim0 u (x) = 0
if x = 0,
Math 4470/5470 Homework #8 Solutions
1. Here is an existence proof for utt c2 uxx = F (x, t) on 0 < t < , < x < , with
initial conditions u(x, 0) = f (x) and ut (x, 0) = g(x). (In class we proved existence for
the bounded version 0 < x < L.)
(a) Show that
Math 4470/5470 Homework #12 Solutions
1. If f (x) =
the formula
n=0
an cos nx for f (x) = sin 2x on 0 x , nd the coecients an using
cos A sin B =
1
2
sin (A + B ) sin (A B ) .
Does the Fourier series converge? How do you know?
Solution: The formula says
a
Math 4470/5470 Homework #3 Solutions
1. Suppose that f (t) is a continuous function on R and that a is some constant. Prove
that any solution of u (t) = f (t)u(t) with u(0) = a is unique, by showing that for any
two solutions u1 (t) and u2 (t), the quotie
Math 4470/5470 Homework #1 Solutions
1. Section 1.0, page 4. Exercises 12. (Specify the most general solution satisfying the
conditions; is it unique?)
Exercise 1: Integrate/solve.
(a) u (x) = 0,
< x <
Solution: u(x) = Ax + B for some constants A and B
Math 4470/5470 Final Exam Solutions
1. (10 points) Presentation questions. (Circle one, no partial credit.)
(a) What does it mean to hear the shape of a drum?
i. To prove well-posedness of the wave equation in a planar domain.
ii. To reconstruct the geome
Math 4470/5470 Exam #1 Solutions
1. (a) (15 points) What are the three conditions that make a linear partial dierential
equation with given initial and/or boundary conditions well-posed ?
Give a one-sentence denition for each of the terms.
Solution:
Exis
Math 4470/5470 Exam #2 Solutions
1. (a) (15 points) What must a and b be for
u ( x ) =
a 0<x<
b otherwise
to be an approximate delta-function? Explain briey and draw a picture.
Solution: We need three conditions: u (x) 0 for all x, lim0 u (x) = 0
if x = 0
Math 4470/5470 Homework #4 Solutions
1. (A linear algebra problem) Suppose A is a symmetric n n matrix; in other words
A = A. Suppose u and v are eigenvectors of A with Au = u and Av = v where
= . Show that u v = u, v = 0. (Hint: rst show that (Au) v = u
Math 4470/5470 Homework #5 Solutions
1. It is not quite true that every eigenvalue problem f (x) = f (x) with homogeneous
boundary conditions has only negative eigenvalues: Robin boundary conditions can
break this.
(a) Show that there is exactly one posit
Math 4470/5470 Homework #10 Solutions
1. Prove the Dini formula
N
sin (N + 1 )
1
2
+
cos n =
.
1
2 n=1
2 sin 2
Hint: using Eulers formula ei = cos + i sin , write cos n = 1 (ein + ein ). Then
2
use the geometric series formula
N
rn =
n=1
rN +1 r
r1
with
Math 4470/5470 Homework #11 Solutions
1. Use the Maximum Principle for the heat equation (from Homework #10) to prove
uniqueness for the problem ut = uxx in the domain 0 x L and t > 0, with
given initial condition u(x, 0) = f (x) and given boundary condit
Math 4470/5470 Homework #8 Solutions
1. Here is an existence proof for utt c2 uxx = F (x, t) on 0 < t < , < x < , with
initial conditions u(x, 0) = f (x) and ut (x, 0) = g (x). (In class we proved existence for
the bounded version 0 < x < L.)
(a) Show tha
Math 4470/5470 Homework #8 Solutions
1. Suppose u(x, y ) is a function satisfying uxx + uyy = 0 (such a function is called harmonic ). A function v (x, y ) satisfying the Cauchy-Riemann equations
v
u
=
x
y
u
v
=
y
x
and
is called a harmonic conjugate of u
Math 4470/5470 Homework #6 Solutions
1. Find the Green function for the problem u (x) + u(x) = 0 with boundary conditions
u (0) = 0 and u (1) = 0. In other words solve the problem
u ( x ) + u( x ) = 0
for 0 < x < 1 with x = y,
u (0) = 0,
u (1) = 0
with li
Math 4470/5470 Homework #7 Solutions
1. Suppose we want to minimize
1
E=
0
e 2x v ( x ) 2 v ( x ) 2 d x
over all functions v satisfying v (0) = 0 and v (1) = 1.
(a) Derive the dierential equation that the minimizing function u(x) will satisfy.
Solution: T
Math 4470/5470 Final Exam Solutions
1. (10 points) Presentation questions. (Circle one, no partial credit.)
(a) What does it mean to hear the shape of a drum?
i. To prove well-posedness of the wave equation in a planar domain.
ii. To reconstruct the geome