Mathematics 217 Exam 2 21 October 2014

Mathematics 217
Exam 2
21 October 2014
Directions: This exam should consist of 18 problems. The first 13 are multiple choice
problems, and the remaining five are hand-graded problems. Multiple choice problems are
worth 5 points apiece. The hand-graded problems are worth 10 points.
Write your student ID number carefully at the top of your answer card, using one line
per digit. Print your name on the top of the card in the space provided. Mark your answer
card with a PENCIL by filling in the appropriate box. On the hand-graded sheets, write
your name and student ID number in the spaces provided.
You may not use a calculator or any written aids.
1
Problem 1: Suppose that P (t) denotes the population of true Justin Bieber fans as a
function of time. Consider the population model
dP
= P (P − 1, 000, 000).
dt
If P (0) = 500, 000, which of the following is a more accurate description of the long-term
behavior of the population of Justin Bieber fans under this model? (Hint: don’t try to integrate and solve for P . Instead, use your understanding of population models and consider
is at t = 0.)
what the sign of dP
dt
A. Eventually, there are no Justin Bieber fans.
B. The population of Bieber fans grows out of control, eventually causing humanity to
collapse.
Problem 2: Consider the following model of free, damped oscillation: y 00 + 4y 0 + y = 0. Is
this model underdamped, overdamped, or critically damped?
A. Underdamped
B. Overdamped
C. Critically damped
2
Problem 3: Let x(t) and v(t) denote the velocity and position of an object. Suppose that
dv
= −v. If both the initial velocity and initial position of the object are positive, which of
dt
the following most accurately describes the long-term motion of the object?
A. The object travels toward infinity and never slows down.
B. The object travels toward infinity, but the speed of the object tends toward zero.
C. The position of the object tends toward some finite value.
Problem 4: Consider the functions x and x2 . Which of the following statements is correct?
(Hint: you probably ought to know whether x and x2 are linearly independent or not.)
A.
B.
C.
D.
x
x
x
x
and
and
and
and
x2
x2
x2
x2
are
are
are
are
linearly
linearly
linearly
linearly
dependent because their Wronskian is zero at x = 0.
independent because their Wronskian is zero at x = 0.
dependent because their Wronskian is nonzero at x = 1.
independent because their Wronskian is nonzero at x = 1.
3
Problem 5: Which of the following is the general solution to the equation y 00 +3y 0 +2y = 0?
Here A and B are arbitrary constants.
A. y = Aex + Be−x
B. y = Aex + Bxex
C. y = Aex + Be−2x
D. y = Ae−2x + Bxe−2x
E. y = Ae−x + Be−2x
F. y = Ae−x + Bxe−x
G. y = Ae3x + Bxe3x
H. y = Ae−3x + Bxe−3x
Problem 6: Which of the following is the general solution to the equation y 00 +6y 0 +9y = 0?
Here A and B are arbitrary constants.
A. y = Aex + Be−x
B. y = Aex + Bxex
C. y = Aex + Be−2x
D. y = Ae−2x + Bxe−2x
E. y = Ae−x + Be−2x
F. y = Ae−x + Bxe−x
G. y = Ae3x + Bxe3x
H. y = Ae−3x + Bxe−3x
4
Problem 7: Suppose we have a linear, homogeneous differential equation with real, constant coefficients. Suppose that this equation is of order m. Suppose that ex , sin x, and
sin 2x are solutions to this equation. What is the smallest value m could be? (Hint: what
are the corresponding factors of the characteristic polynomial?)
A.
B.
C.
D.
E.
F.
G.
H.
0
1
2
3
4
5
6
7
Problem 8: Suppose that p, q, and f are functions which are continuous on (−∞, ∞).
If y1 and y2 are linearly independent solutions to y 00 + p(x)y 0 + q(x) = f (x), which of the
following values can the Wronskian, W (y1 , y2 ), not take?
A.
B.
C.
D.
E.
F.
G.
H.
0
1
2
3
4
5
6
7
5
Problem 9: Suppose that a constant coefficient, homogeneous differential equation has
characteristic equation
(r + 2)3 (r − 2i)2 (r + 2i)2 = 0.
Which of the following is the general solution to this problem? All upper-case letters are
arbitrary constants.
A.
B.
C.
D.
E.
F.
G.
H.
y
y
y
y
y
y
y
y
= (A + Bx + Cx2 )e−2x + (D + Ex) cos 2x + (F + Gx) sin 2x
= (A + Bx + Cx2 )e−2x + D cos 2x + F sin 2x
= (A + Bx + Cx2 )e−2x + D cos2 2x + F sin2 2x
= (A + Bx + Cx2 )e−2x + D cos2 4x + F sin2 4x
= (A + Bx + Cx2 + Dx3 )e−2x + (E + F x) cos 2x + (G + Hx) sin 2x
= (A + Bx + Cx2 + Dx3 )e−2x + E cos 2x + G sin 2x
= (A + Bx + Cx2 + Dx3 )e−2x + E cos2 2x + G sin2 2x
= (A + Bx + Cx2 + Dx3 )e−2x + E cos2 4x + G sin2 4x
Problem 10: Consider the equation y 00 + y = x2 . Which of the following is the general
solution to this problem? A and B are arbitrary constants.
A.
B.
C.
D.
E.
F.
G.
H.
y
y
y
y
y
y
y
y
= A cos x + B sin x + x2 + x + 1
= A cos x + B sin x + x2 + x − 1
= A cos x + B sin x + x2 + x + 2
= A cos x + B sin x + x2 + x − 2
= A cos x + B sin x + x2 + 1
= A cos x + B sin x + x2 − 1
= A cos x + B sin x + x2 + 2
= A cos x + B sin x + x2 − 2
6
3 d
d
Problem 11: Consider the equation dx
+ 1 ( dx
+ 2)4 y = x2 e−2x . Using the Method of
Undetermined Coefficients, which of the following should we guess for yp ? Here upper-case
letters denote arbitrary constants.
A.
B.
C.
D.
E.
F.
G.
H.
yp
yp
yp
yp
yp
yp
yp
yp
= Ax2 e−2x
= (A + Bx)e−2x
= (A + Bx + Cx2 )e−2x
= (A + Bx + Cx2 + Dx3 )e−2x
= (Ax + Bx2 + Cx3 )e−2x
= (Ax2 + Bx3 + Cx4 )e−2x
= (Ax3 + Bx4 + Cx5 )e−2x
= (Ax4 + Bx5 + Cx6 )e−2x
Problem
the equation y 0 + xy = x2 with initial value y(0) = 1. If we plug
P∞12: Consider
n
in y = n=0 an x , what value must a3 be?
A.
B.
C.
D.
E.
F.
G.
H.
a3 = −1
a3 = −2/3
a3 = −1/3
a3 = 0
a3 = 1/3
a3 = 2/3
a3 = 1
None of the above.
7
Problem 13: Consider the following model of forced vibrations without friction: y 00 + y =
−2 cos(ωt). You know that resonance occurs in this model. What is the resonance frequency?
A.
B.
C.
D.
E.
F.
G.
H.
ω = −3
ω = −2
ω = −1
ω=0
ω=1
ω=2
ω=3
I can’t answer this question because I’m too upset about what you said about Justin
Bieber before.
8
Math 217 Exam 2: Hand-Graded Problems
Student Name:
Student ID #:
Problem 14:
Consider the equation y 000 − 8y = e2x .
1. Write down the general solution, yh , of the homogeneous equation, y 000 − 8y = 0. (Hint:
2 is a root of the characteristic polynomial.)
2. In the Method of Undetermined Coefficients, what should our guess be for yp ?
3. Determine yp exactly and write down the general form for solutions to the equation,
y 000 − 8y = e2x .
9
Math 217 Exam 2: Hand-Graded Problems
Student Name:
Student ID #:
Problem 15: Consider the equation (2x − 1)y 0 + 2y = 0 with initial value y(0) = 1.
P
n
1. If y = ∞
n=0 an x , compute a0 through a5 .
an
2. Determine the ratio an+1
. (Just computing this ratio for the first few terms is not
enough. You need to find the general equation relating an and an+1 and compute their
ratio using that equation.) What is the radius of convergence for this series?
3. (Bonus). What rational function has this power series? (You will get full credit if you
don’t answer this part of the problem.)
10
Math 217 Exam 2: Hand-Graded Problems
Student Name:
Student ID #:
Problem 16:
semester?
What is the funniest thing that you have experienced, seen, or heard this
11
Math 217 Exam 2: Hand-Graded Problems
Student Name:
Student ID #:
Problem 17: Consider the equation
y 00 + y = cos x.
1. Write down the general form of solutions to the homogeneous equation y 00 + y = 0.
2. What guess for yp should we use in the Method of Undetermined Coefficients?
3. Use your guess to determine exactly what yp is.
4. Write the formula for the general solution to y 00 + y = cos x.
12
Math 217 Exam 2: Hand-Graded Problems
Student Name:
Student ID #:
Problem 18: Consider the equation y 00 − x4 y 0 +
1. Consider the equation y 00 − x4 y 0 +
is a solution to this equation.
6
y
x2
6
y
x2
= x3 .
= 0. Determine the two values of p for which xp
2. Using the method of variation of parameters, determine a particular solution yp for
this equation.
3. Write down the general solution to this equation (on the interval (0, ∞)).
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