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MATH 251
Final Examination
May 3, 2010
FORM A
Name: Student Number: Section:
This exam has 16 questions for a total of 150 points. In order to obtain full credit for partial credit problems, all work must be shown. For other problems, points might be deducted, at the sole discretion of the instructor, for an answer not supported by a reasonable amount of work. The point value for each question is in parentheses to the right of the question number. A table of Laplace transforms is attached as the last page of the exam.
Please turn off and put away your cell phone.
You may not use a calculator on this exam.
thru 11:
12: 13: 14: 15: 16:
Total:
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- (6 points) Consider the autonomous equation
y′^ = 4y^2 − y^4.
Suppose y(99) = λ and lim t→∞ y(t) = 2, find all possible value(s) of λ.
(a) − 2 < λ < ∞
(b) λ = 2
(c) 0 < λ < ∞
(d) 0 < λ < 2
- (6 points) Consider the two problems below.
(I) y′′^ + λy = 0, y(0) = 2π, y′(0) = β.
(II) y′′^ + λy = 0, y′(0) = 0, y′(2π) = β.
(a) Only (I) has a unique solution for every combination of real numbers λ and β.
(b) Only (II) has a unique solution for every combination of real numbers λ and β.
(c) Each has a unique solution for every combination of real numbers λ and β.
(d) Neither is guaranteed to have a unique solution for every combination of real numbers λ and β.
- (6 points) Find the inverse Laplace transform of F (s) = e−^11 s^
(s + 1)^3
(a) u 11 (t)t^2 et
(b) δ(t − 11)t^2 e−t
(c) u 11 (t)(t − 11)^2 e−t+
(d) δ(t − 11)t^2 e−t−^11
- (6 points) Find the solution of the initial value problem
y′′^ + 9y = δ(t − 3), y(0) = 1, y′(0) = 0.
(a) y = − cos(3t) +
δ(t − 2) sin(3t)
(b) y = cos(3t) +
u 3 (t) sin(3t − 9)
(c) y = − cos(3t) + u 3 (t) sin(3t)
(d) y = cos(3t) +
u 3 (t) sin(3t + 9)
- (6 points) Suppose the linear system below has an asymptotically stable improper node at (0, 0). What is/are the value(s) of α?
x′^ =
[
− 3 −α − 1 α
]
x.
(a) 1
(b) − 1
(c) 1, 9
(d) − 1 , − 9
- (6 points) Given that the point (0, 1) is a critical point of the nonlinear system of equations
x′^ = xy^2 − 2 xy y′^ = xy + y − x − 1
This critical point (0, 1) is a(n)
(a) asymptotically stable spiral point.
(b) (neutrally) stable center.
(c) unstable saddle point.
(d) asymptotically stable improper node.
- (6 points) Consider the wave equation initial-boundary value problem
9 uxx = utt, 0 < x < 2 , t > 0 u(0, t) = 0, u(2, t) = 0, u(x, 0) = h(x), ut(x, 0) = 0.
In what specific form will its general solution appear?
(a) u(x, t) =
∑^ ∞
n=
An cos 9 nπt 2
sin nπx 2
(b) u(x, t) =
∑^ ∞
n=
An cos 3 nπt 2
sin nπx 2
(c) u(x, t) =
∑^ ∞
n=
Bn sin 3 nπt 2
sin nπx 2
(d) u(x, t) =
∑^ ∞
n=
Bn sin
9 nπt 2 sin
nπx 2
- (18 points) Consider the list of differential equations below.
A. y′′^ + 6y′^ + 5y = 0
B. y′′^ + 25y = 0
C. t y′^ − 2 t y = e−^3 t^ sin(t)
D. y′′^ + 4y′^ + 5y = 0
E. y′′^ − 2 y′^ + y = 0
F. y(4)^ + 2y′′^ + y = 0
G. y′^ + 2y = t−^9
For each part, write down the letter corresponding to the equation on the list with the specified properties. There is only one correct answer to each part, but the same letter could be used more than once for different parts.
(a) (3 points) This first order equation can be solved using the integrating factor μ(t) = e−^2 t.
(b) (3 points) This equation describes a mass-spring system without damping.
(c) (3 points) This equation describes an overdamped mass-spring system.
(d) (3 points) This equation has y = 2t et^ as a solution.
(e) (3 points) Every solution of this equation is periodic.
(f) (3 points) This equation has y = C 1 cos(t)+C 2 sin(t)+C 3 t cos(t)+C 4 t sin(t) as its general solution.
- (16 points) Consider the two-point boundary value problem
X′′^ + λX = 0, X′(0) = 0, X′(10) = 0.
(a) (12 points) Find all positive eigenvalues and corresponding eigenfunctions of the bound- ary value problem.
(b) (4 points) Is λ = 0 an eigenvalue of this problem? If yes, find its corresponding eigenfunc- tion. If no, briefly explain why it is not an eigenvalue.
- (16 points) Let f (x) = 1 − x^2 , 0 < x < 1.
(a) (4 points) Consider the odd periodic extension, of period T = 2, of f (x). Sketch 3 periods, on the interval − 3 < x < 3, of this odd periodic extension.
(b) (4 points) Find a 1 , the first cosine coefficient of the Fourier series of the periodic function described in (a).
(c) (4 points) Which of the integrals below can be used to find the Fourier sine coefficients of the odd periodic extension in (a)?
(i) bn =
0
(1 − x^2 ) sin nπx 2
dx
(ii) bn =
− 1
(1 − x^2 ) sin nπx 2
dx
(iii) bn = 2
0
(1 − x^2 ) sin(nπx) dx
(iv) bn = 4
− 1
(1 − x^2 ) sin(nπx) dx
(d) (4 points) To what value does the Fourier series of this odd periodic extension converge at x = −2? At x =
