APPLIED MATHEMATICS
MASTER’S EXAM
Fall 2008
1
Ordinary Differential Equations
1. Consider the boundary value problem
y 00(x) + y(x) = sin2(αx),
1
y(0) = y(π) = .
2
For which values of α does the problem have a solution ? Find the
solution when it exists. Is the solution in unique ?
2. Consider the system of ODEs
x0(t) = 3x(t) + y(t) + 1,
y 0(t) = x(t) + 3y(t) + t.
(a) Find the general solution.
(b) Find the solution that satisfies x(0) = 1, y(0) = 0.
3. Consider the autonomous system :
x0(t) = x(t) − x(t)y(t),
y 0(t) = x(t)y(t) − y(t).
(a) Find all equilibrium points in the phase (x, y) plane.
(b) Classify each as to type (i.e., saddle, center, spiral, node, etc.).
(c) Determine the linear and nonlinear stability of each.
(d) Sketch a rough phase plane portrait, in the range x ≥ 0, y ≥ 0.
APPLIED MATHEMATICS
MASTER’S EXAM
Fall 2008
2
Partial Differential Equations
4. Consider the heat equation
ut = uxx − sin(πx) + sin(3πx),
u(0, t) = u(1, t) = 0,
u(x, 0) = sin(πx).
(a) Solve this nonhomogeneous problem.
(b) What is the limit of u(x, t) as t → ∞ ?
5. Solve the following Dirichlet problem
1
1
∆u = urr + ur + 2 uθθ = −1,
r
r
u(1, θ) = sin(θ).
r<1
Assume also that u is bounded at r = 0.
6. Solve the one-dimensional wave equation
utt = uxx;
−∞ < x < ∞, t > 0
u(x, 0) = exp(−x2), ut(x, 0) = 0.
Now consider the finite interval problem where 0 < x < 1 and we
have the additional boundary conditions u(0, t) = 0 = u(1, t), for
t > 0. In what portion of the space time strip {0 < x < 1, t > 0}
does the solution to the finite interval problem coincide with that of
the infinite interval problem you solved above ? You need not solve
the finite interval problem completely.
APPLIED MATHEMATICS
MASTER’S EXAM
Fall 2008
3
Complex Variables
7. Consider the function
f (z) =
1
.
(1 + z)2 (3 − z)
Find all singular points, and compute the residues at these points.
R
Compute the value of the contour integral
if C is the
C f (z)dz
following circle (taken counterclockwise):
(a)
|z| = 1/2
(b)
|z| = 2
(c)
|z| = 2008.
8. Compute the Taylor or Laurent series of the functions
(a)
sin (ez − 1)
(b)
1+z
sin(z)
about z = 0. If it is a Laurent series about z0 find the coefficients of
(z − z0)N for N = −2, −1, 0, 1, 2 (some of these coefficients may be
zero). For Taylor series give the first three non-zero terms. Give also
the domain where the series’ converge.
APPLIED MATHEMATICS
MASTER’S EXAM
Fall 2008
4
Linear Algebra
9. Consider the 3 × 3 matrix


A=

0 1 −1
1 0
−1 2


2
.
0
(a) Find all eigenvalues and eigenvectors of A.
(b) Find all eigenvalues and eigenvectors of A−1 .
10. Let V be a subspace of R4 spanned by the vectors








1
0
−2
1








 0
 −1 
 1
 0








v1 = 
 , v2 = 
 , v3 = 
 , v4 = 
.
 1
 −2 
 3
 −2 








−1
2
0
−1
(a) Find a basis for V .
(b) Find a basis for V ⊥, the orthogonal complement of V .
(c) Use the Gram-Schmit method to compute an orthonormal basis
for V .
(d) Extend the orthonormal basis from (c) to a basis for R4.
APPLIED MATHEMATICS
MASTER’S EXAM
Fall 2008
5
Advanced Calculus
11. Consider the following infinite series:
(a)
∞
X
n=1
ln(1/n)
n
,
(b)
∞
X
cos(πn2 )
n=2
ln n
,
(c)
∞
X
n=2
1
,
n lnα n
∞ X
1
1
+
+ nβ .
(d)
tan−1 ln 1 − √
n
2n
n=1
For (a) and (b) determine whether they converge, and whether they
converge absolutely. For (c) and (d) find the (real) values of α and
β for which the series converge.
12. Consider the ellipse x2 + xy + y 2 = 1 and the line x + y = 2 .
(a) Determine the points on the ellipse with maximal and minimal
distance to the line. Give both the location of the points and the
maximal and minimal distances.
(b) Determine the points of the ellipse where tangents to the ellipse
are orthogonal to the line.