Quiz VI, Math 403, 17 November, 2010
Name:
Solutions
1. (30 points:) Solve the problem
2u
t2
u(0, y, t)
u(x, 0, t)
u(x, y, 0)
u
(x, y, 0)
t
2u 2u
, 0<x< , 0<y< ,
+
x2 y 2
= u(, y, t) = 0 , 0 y , t > 0 ,
= u(x, , t) = 0 , 0 x , t > 0 ,
= f (x, y ) , 0 x
Quiz V, Math 403, 3 November, 2010
Name:
Solutions
1. (30 points:) Let h > 0 be constant. Solve the problem
u
2u
= k 2 hu , 0 < x < L ,
t
x
u
u
(0, t) =
(L, t) = 0 , t > 0 ,
x
x
u(x, 0) = f (x) , 0 < x < L
t>0,
using separation of variables.
Solution: Su
Quiz IV, Math 403, 20 October, 2010
Name:
Solutions
1. (15 points:) Show that the set cfw_sin(nx), n = 1, 2, 3, . . ., is orthogonal on the
interval [0, ].
Solution: Suppose that m and n are distinct positive integers. We must show that
I :=
sin(mx) sin(n
Quiz III, Math 403, 15 September, 2010
Name:
Solutions
1. (30 points:) Find the eigenvalues and eigenvectors of the matrix
1 2
2
1
A=
Solution: Consider
.
1
2
2 1
A I =
.
Then the eigenvalues of A are the roots of characteristic polynomial:
p() =
=
=
=
=
Quiz II, Math 403, 15 September, 2010
Name:
Solutions
1. (10 points:) Use the Gram-Schmidt Process to convert B = cfw_u1 , u2 , u3 into
B = cfw_v1 , v2 , v3 , where B is an orthogonal set. Use the vectors
4
1
0
0
2
0
u1 =
6 , u2 = 6 , u3 = 1 .
1
0
1
Quiz I, Math 403, 1 September, 2010
Name:
Solutions
1. (10 points:) Let V = P3 , where P3 is the vector space (VS) of all polynomials of degree less than or equal to 3 and let W = cfw_f P3 | f (x) = c3 x3 + c1 x, c3 , c1 R.
(In words, W is the set of all
Practice Midterm, Math 403
27 September, 2010
Name:
Version 1
1. Problems 11, 13, and 21, Section 8.10, page 406.
2. Problems 9, 11, 23, and 25, Section 8.8, page 395.
3. Given that the eigenvalues of the matrix
7
4 4
A = 4 8 1
4 1 8
are 1 = 2 = 9 and 3
Practice Final, Math 403
24 November, 2010
Name:
Version 4
1. Show that
nx
p
sin
, n = 1, 2, 3, . . .
is orthogonal on the interval [0, p]. Find the norm of each function.
2. Show that
nx
p
1, cos
, n = 1, 2, 3, . . .
is orthogonal on the interval [0, p].
Midterm Exam, Math 403, 4 October, 2010
Name:
Solutions
1. (10 points:) Consider the set B = cfw_u1 , u2 , where u1 , u2 R3 are given below.
Find an orthonormal set of vectors from R3 , call it B , such that span(B ) =
span(B ).
1
0
u1 = 0 , u2 = 2 .
1
1
Final Exam, Math 403
2 December, 2010
Name:
Solutions
1. (10 points): Compute the Fourier series of f (x) = x on [1, 1]. Then, without
doing any extra work, write down the Fourier Sine series of f (x) = x on [0, 1].
Solution: We wish to expand the functio