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Math 240 Final Exam, spring 2011
Name (printed):
TA:
Recitation Time:
This examination consists of ten (10 problems). Please turn off all electronic
devices. You may use both sides of a 8. 5 × 11 sheet of paper for notes while
you take this exam. No calculators, no course notes, no books, no help from
your neighbors. Show all work, even on multiple choice or short answer
questions—the grading will be bases on your work shown as well as the end
result. Please fill in your final answer in the underlined space in each
problem. Remember to put your name at the top of this page. Good luck.
My signature below certifies that I have complied with the Univer-
sity of Pennsylvania’s code of academic integrity in completing this
examination.
Your signature
Problem Score (out of )
1 (10)
2 (10)
3 (10)
4 (10)
5 (10)
6 (10)
7 (10)
8 (10)
9 (10)
10 (10)
Total (100)
- (10 pts) (a) Give an example of a 3 × 3 matrix A which has only two distinct eigenvalues and A is not diagonalizable. In other words, there does not exist an invertible 3 × 3 matrix C such that C−^1 · A · C is a diagonal matrix. Justify your answer.
A =
(b) Give an example of a 3 × 3 matrix B which has only two distinct eigenvalues and B is diagonalizable. Justify your answer.
B =
- (10 pts) Give an example of a homogeneous linear ordinary differential equation which is not ordinary at x = 0 and has a regular singular point at x = 0, which has two linearly independent solutions of the form
xμi^ ·
m≥ 1
amxm
, i = 1, 2 , μ 1 , μ 2 ∈ C, μ 1 6 = μ 2
for two distinct complex numbers μ 1 and μ 2. Solve the differential equation you create. (Hint: The two numbers μ 1 , μ 2 in some of the easier examples are integers.)
The differential equation is.
The two linearly independent solutions are.
- (10 pts) Find the general solution to the following differential equation
y′′^ − 5 y′^ + 6y = e^2 x
(Your answer should involve some unspecified constants.)
y =.
(b) Evaluate these two line integrals.
C 1
(x − y) dx + (x + y) dy x^2 + y^2
C 2
(x − y) dx + (x + y) dy x^2 + y^2
- (10 pts) Let S = ∂D be the boundary of the solid region D contained in the cylinder x^2 + y^2 = 4 between z = x and z = 8, i.e.
D = {(x, y, z) ∈ R^3 | x^2 + y^2 ≤ 4 , x ≤ z ≤ 8 }.
Let n be the unit normal vector field on S pointing outward relative to D. Calculate the flux (^) ∫ ∫
S
F · n dS
of the vector field F = 〈x, y^2 , z + y〉 = x~i + y^2 ~j + (z + y) ~k.
S
F · n dS =.
- (10 pts) Find a recursion formula for the coefficients an’s of a power series expansion of a function
y(x) = 1 +
∑^ ∞
n=
an xn
defined on (− 1 , 1) which satisfies the following differential equation
(x − 1)
d^2 y dx^2
dy dx
Recursion formula:.
- (10 pts) Find the general solution to the system of linear ordinary differential equation
d dx
u(x) =
u(x),^ where^ u(x) =
u 1 (x) u 2 (x) u 3 (x)
(Your answer should involve some unspecified constants.)
u(x) =
C. The differential equation x d
(^2) y dx^2 + (cos(x)^ −^ 1)^
dy dx = 0 for the function^ y(x) on R is equivalent to a linear ordinary differential equation for y(x) on R which has no singular point.
D. Suppose that f 0 (x), f 1 (x), and f 2 (x) are polynomials in x, and f 2 (x) is not identically 0. If y 1 (x) and y 2 (x) are smooth functions (i.e. they can be differentiated infinitely many times) on R such that y 1 (0) = y 2 (0), y′ 1 (0) = y′ 2 (0), and
f 2 (x)y′′ i + f 1 (x)y′ i + f 0 (x)yi = 0 for i = 1, 2 ,
then y 1 (x) = y 2 (x).
E. Suppose that A is a 2 × 2 matrix with real entries, and x(t), y(t) are two differentiable functions on R such that
d dt
[
x(t) y(t)
]
= A ·
[
x(t) y(t)
]
and x(t)^2 + y(t)^2 = 1
for all real numbers t ∈ R. Then the two eigenvalues λ 1 , λ 2 of A has both purely imaginary numbers, i.e. λ 1 , λ 2 ∈
− 1 · R.
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