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MATH 251
Examination II
November 13, 2012
FORM A
Name: Student Number: Section:
This exam has 12 questions for a total of 100 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.
through 8:
9:
10:
11:
12:
Total:
Do not write in this box.
- (5 points) Consider the fourth order linear equation
y(4)^ − 2 y(3)^ + y′′^ = 0.
What is its general solution?
(a) y(t) = C 1 + C 2 t + C 3 et^ + C 4 tet
(b) y(t) = C 1 et^ + C 2 e−t^ + C 3 cos t + C 4 sin t
(c) y(t) = C 1 + C 2 t + C 3 cos t + C 4 sin t
(d) y(t) = C 1 + C 2 t + C 3 e−t^ + C 4 te−t
- (5 points) Evaluate the following definite integral ∫ (^) ∞
0
e−(s+4)tt^2 dt.
(a)
(s − 4)^3
(b) e−s^
(s − 4)^3
(c) e−s^
(s + 4)^3
(d)
(s + 4)^3
- (5 points) Which system of first order linear equations below is equivalent to the second order linear equation y′′^ − 4 y′^ + 7y = 10t^3?
(a)
x′ 1 = x 2 x′ 2 = − 7 x 1 + 4x 2 + 10t^3
(b)
x′ 1 = x 2 x′ 2 = 7 x 1 − 4 x 2 + 10t^3
(c)
x′ 1 = x 2 x′ 2 = − 4 x 1 + 7x 2 + 10t^3
(d)
x′ 1 = x 2 x′ 2 = 4 x 1 − 7 x 2 + 10t^3
- (5 points) Consider a certain system of two first order linear differential equations in two unknowns, x′^ = Ax, where A is a matrix of real numbers. Suppose one of the eigenvalues of the coefficient matrix A is r = 3 + i, which has a corresponding eigenvector
[
− 2 − 5 i 4
]
. What is the system’s real-valued general solution?
(a) x(t) = C 1 e^3 t
[
−2 cos t − 5 sin t 4 cos t
]
[
5 cos t − 2 sin t 4 sin t
]
(b) x(t) = C 1 e^3 t
[
−2 cos t + 5 sin t 4 cos t
]
[
−5 cos t − 2 sin t 4 sin t
]
(c) x(t) = C 1 e^3 t
[
−2 cos t + 5 sin t 4 sin t
]
[
−5 cos t − 2 sin t −4 cos t
]
(d) x(t) = C 1 e^3 t
[
−2 cos t − 5 sin t 4 sin t
]
[
5 cos t − 2 sin t −4 cos t
]
- (5 points) For what value(s) of β will the linear system below have a (neutrally) stable center at (0, 0)? x′^ =
[
2 β 5 − 5 β^2
]
x.
(a) 0
(b) 2
(c) − 2 , 2
(d) 0, − 2
- (5 points) Given that (− 2 , 4) is a critical point of the nonlinear system
x′^ = xy + 2y y′^ = xy − 4 x
Linearize this system about (− 2 , 4) to determine the critical point as a(n)
(a) unstable saddle point.
(b) asymptotically stable spiral point.
(c) unstable node.
(d) asymptotically stable proper node.
- (15 points) Find the inverse Laplace transform of each function given below.
(a) (7 points) F (s) =
2 s + 5 s^2 (s − 2)
(b) (8 points) F (s) = e−^4 s^
s s^2 + 8s + 20 − 4 e−^7 s
- (18 points)
(a) (16 points) Use the Laplace transform to solve the following initial value problem.
y′′^ + 3y′^ + 2y = u 6 (t) − 2 δ(t − 1), y(0) = 0, y′(0) = 2.
(b) (2 points) Evaluate y(ln 2).
