Tufts University
Department of Mathematics
Math 250-03 Homework 1
Due: Thursday, September 20, at 3:00 p.m. (in class).
1. Solve the following ODEs. You may use any method you like, but must show your work; you
may use any textbooks you like, but must do
Tufts University
Department of Mathematics
Math 250-03 Homework 3
Due: Thursday, October 4, at 3:00 p.m. (in class).
1. (20 points) Find an explicit series solution for
ut = uxx + et sin(3x) 0 < x < 1, t > 0
u(0, t) = u(1, t) = 0
t0
.
u(x, 0) = x sin(x)
Tufts University
Department of Mathematics
Math 250-03 Homework 4
Due: Thursday, October 11, at 3:00 p.m. (in class).
1. (20 points) Recall the denition of a continuous function: The function f : [a, b] R
is continuous at x0 if, for any , there exists a ,
Tufts University
Department of Mathematics
Math 250-03 Homework 3
Due: Thursday, October 4, at 3:00 p.m. (in class).
1. (20 points) Find an explicit series solution for
ut = uxx + et sin(3x) 0 < x < 1, t > 0
u(0, t) = u(1, t) = 0
t0
.
u(x, 0) = x sin(x)
Tufts University
Department of Mathematics
Math 250-03 Homework 2
Due: Thursday, September 27, at 3:00 p.m. (in class).
1. (20 points) Find an explicit series solution for
0 < x < 1, t > 0
ut = Kuxx
u(0, t) = u(1, t) = 0 t 0
.
u(x, 0) = x(1 x)
0x1
Soluti
Tufts University
Department of Mathematics
Math 250-03 Homework 1
Due: Thursday, September 20, at 3:00 p.m. (in class).
1. Solve the following ODEs. You may use any method you like, but must show your work; you
may use any textbooks you like, but must do
Tufts University
Department of Mathematics
Math 250-03 Homework 2
Due: Thursday, September 27, at 3:00 p.m. (in class).
1. (20 points) Find an explicit series solution for
0 < x < 1, t > 0
ut = Kuxx
u(0, t) = u(1, t) = 0 t 0
.
u(x, 0) = x(1 x)
0x1
2. (20
Tufts University
Department of Mathematics
Math 250-03 Homework 4
Due: Thursday, October 11, at 3:00 p.m. (in class).
1. (20 points) Recall the denition of a continuous function: The function f : [a, b] R
is continuous at x0 if, for any , there exists a ,
Tufts University
Department of Mathematics
Math 250-03 Homework 5
Due: Thursday, November 1, at 3:00 p.m. (in class).
1. (10 points) Let A B C be normed spaces with common norm
dense in B and B is dense in C , then A is dense in C .
. Prove that if A is
Tufts University
Department of Mathematics
Math 250-03 Take-Home Midterm Exam
Due: Thursday, October 25, at 3:00 p.m. (in class). No extensions!
1. (15 points) Let K (x) be a bounded positive function for x [0, 1]. Solve
x (K (x)x u(x) = 0 for 0 < x < 1,
Tufts University
Department of Mathematics
Math 250-03 Homework 8
Due: Thursday, November 29, at 3:00 p.m. (in class).
1. (15 points) Let (x) be the Dirac delta distribution, and
Denition 1.26. Show that
0
n (m)
(1)n n! (x)
x (x) =
(1)n m! (mn)
(x)
(mn)!
Tufts University
Department of Mathematics
Math 250-03 Homework 7
Due: Thursday, November 15, at 3:00 p.m. (in class).
1. (10 points) Let H be a Hilbert Space, and A : H H and B : H H be operators with
adjoints A and B . Prove that
(a) (A + B ) = A + B
(
Tufts University
Department of Mathematics
Math 250-03 Homework 5
Due: Thursday, November 1, at 3:00 p.m. (in class).
1. (10 points) Let A B C be normed spaces with common norm
dense in B and B is dense in C , then A is dense in C .
. Prove that if A is
Tufts University
Department of Mathematics
Math 250-03 Homework 6
Due: Thursday, November 8, at 3:00 p.m. (in class).
1. (20 points) Prove the following Poincar-Friedrichs Theorem.
e
Dene H1 ([a, b]2 ) = u : [a, b]2 R | u, ux , uy L2 ([a, b]2 ) , where L2
Tufts University
Department of Mathematics
Math 250-03 Homework 6
Due: Thursday, November 8, at 3:00 p.m. (in class).
1. (20 points) Prove the following Poincar-Friedrichs Theorem.
e
Dene H1 ([a, b]2 ) = u : [a, b]2 R | u, ux , uy L2 ([a, b]2 ) , where L2