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MATH 251
Examination II
July 28, 2008
Name: Student Number: Section:
This exam has 9 questions for a total of 100 points. In order to obtain full credit for partial credit problems, all work must be shown. Credit will not be given for an answer not supported by 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.
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Total:
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- (5 points) Which system of first order linear equations below is equivalent to the second order linear equation y′′^ + 5y′^ − 6 y = 0?
(a)
x′ 1 = x 2 x′ 2 = − 5 x 1 + 6x 2
(b)
x′ 1 = x 2 x′ 2 = 6 x 1 − 5 x 2
(c)
x′ 1 = −x 1 x′ 2 = 6 x 1 + 5x 2
(d)
x′ 1 = 5 x 1 − 6 x 2 x′ 2 = x 2
- (5 points) Suppose y(t) is the solution of the first order linear initial value problem
y′^ + 2y = t^3 e−^4 t, y(0) = − 3.
What is Y (s), the Laplace transform of y(t)?
(a) Y (s) =
(s + 2)(s + 4)^4
s + 2
(b) Y (s) =
(s + 2)(s + 4)^4
s + 2
(c) Y (s) =
(s + 2)(s − 4)^4
s + 2
(d) Y (s) =
(s + 2)(s − 4)^4
s + 2
- (12 points) Find the inverse Laplace transforms of
(a) (6 points) 3 s^2 − 2 s + 8 s^3 + 4s
(b) (6 points) e−^3 s^ 4 s + 6 s^2 − 6 s + 25
- (12 points) Rewrite the following piecewise continuous function f (t) in terms of the unit-step functions. Then find its Laplace transform L{f (t)}.
f (t) =
1 + 2t^2 , 0 ≤ t < 5 e−^4 t^ − t, t ≥ 5
- (14 points)
(a) (12 points) Solve the initial value problem
x′^ =
[
]
x, x(0) =
[
]
(b) (2 points) Classify the type and stability of the critical point of this system at (0, 0).
- (12 points) In each part below, consider a certain system of two first order linear differential equations in two unknowns, x′^ = Ax. (a) (4 points) Suppose one of the eigenvalues of the coefficient matrix A is r = 4i, which has a corresponding eigenvector
1 − 3 i 2
. Write down the system’s real-valued general solution.
(b) (2 points) Classify the type and stability of the critical point at (0, 0) for the system described in (a).
(c) (4 points) Suppose the coefficient matrix A only has one distinct eigenvalue, r = −7, which has corresponding eigenvectors both
and
. Write down the system’s general solution.
(d) (2 points) Classify the type and stability of the critical point at (0, 0) for the system described in (c).
