APPLIED MATHEMATICS
MASTER’S EXAM
Spring 2008
1
Ordinary Differential Equations
1. Consider the boundary value problem
y 00 (x) + y(x) = x + A,
y(0) = B,
0<x<π
y(π) = C.
(a) For what value(s) of A, B, C does the problem have a solution ?
(b) Find the solution when it exists.
(c) Is the solution in (b) unique ?
2. Consider the ODE
y 00 (x) + (1 + x)y 0 (x) + xy(x) = 0.
(a) Given one nonzero solution y1(x) find a second solution y2 (x) in terms of
y1. This is the method of reduction of order.
(b) Find one solution explicitly (hint : either guess a solution or use Taylor
series).
(c) Find the second linearly independent solution, e.g., by using the formula
in (a).
3. Consider the conservative ODE :
x00 (t) + sin[πx(t)] = 0.
(a) Find all equilibrium points in the phase (x, x0 = y) plane.
(b) Classify each as to type (i.e., saddle, center, spiral, node, etc.). It may
help to use the periodicity of the sine function.
(c) Determine the linear and nonlinear stability of each.
(d) Sketch a rough phase plane portrait.
APPLIED MATHEMATICS
MASTER’S EXAM
Spring 2008
2
Partial Differential Equations
4. Solve the following exterior Dirichlet problem
1
1
∆u = urr + ur + 2 uθθ = 0,
r
r
u(1, θ) = 1 + cos(θ).
r>1
Assume also that u is bounded as r → ∞.
5. Consider the heat equation
ut = uxx + sin(πx);
u(0, t) = 0,
0 < x < 1, t > 0
u(1, t) = 0
u(x, 0) = sin(2πx).
(a) Solve this non homogeneous problem.
(b) What is the limit of u(x, t) as t → ∞ ?
6. Consider the one-dimensional wave equation
utt = c2 uxx ;
−∞ < x < ∞, t > 0
u(x, 0) = f (x), ut (x, 0) = g(x).
Find the general solution using any method. You may assume that the functions f and g have sufficient decay as |x| → ∞.
APPLIED MATHEMATICS
MASTER’S EXAM
Spring 2008
3
Complex Variables
7. Consider the function
f (z) =
1
.
1 − z − z2 + z3
Find all singular points, and compute the residues at these points.
R
Compute the value of the contour integral
f (z)dz if C is the following
C
circle (taken counterclockwise):
(a)
|z| = 1/2
(b)
|z| = 2
(c)
|z + 1| = 1.
8. Compute the Taylor or Laurent series of the function
z
f (z) = sin
z+1
about (a) z = 0 and (b) z = −1. If it is a Laurent series about z0 find the
coefficients of (z − z0 )N for N = −2, −1, 0, 1, 2 (some of the coefficients may
be zero). For Taylor series give the first three non-zero terms.
Linear Algebra
9. Consider the 3 × 3 matrix


1 −1 0



A=
 0 1 −1 .
−1 0 1
(a) Find all eigenvalues and eigenvectors.
(b) Is A diagonalizable over the complex numbers C?
APPLIED MATHEMATICS
MASTER’S EXAM
Spring 2008
4
10. Consider the overdetermined system
2x + y = 1,
x + y = 1,
x + 2y = 1.
(a) Write the system as Ax = b where A is a 3 × 2 matrix and b is a column
vector.
(b) Find the least squares approximate solution to the system.
(c) Compute the error ||Ax∗ − b||, where x∗ = (x∗ , y∗ )T is the approximate
solution and || · || is the standard norm.
(d) Find the projection of b onto the column space of A.
Advanced Calculus
11. Consider the following infinite series’. For (a) and (b) determine if they
converge, and if they converge absolutely. For (c) and (d) determine the
(real) values of β and α for which the series converge.
(a)
∞
X
n=16
(c)
∞
X
n=1
√ √
∞
X
2n + 1 − 2 n n + 1
√
(b)
n
n=1
(−1)n+1
ln ln ln(n)
2
exp[−β ln (n)]
(d)
∞
X
exp[−α ln(n)]
n=1
12. Find the maximum and minimum distances from the origin, for points on
the ellipse
x2 + 2βxy + y 2 = 1,
0 < β < 1.
Give the values of the maximal and minimal distance as well as the points
where they are achieved. Comment on the cases β = 0 and β = 1.