Differential Equations
SHOW ALL WORK!!
Take-Home Exam
NAME:
Instructions: SHOW ALL WORK !! Exam is due at beginning of class on Tuesday 12/11. Consult
no other person regarding these questions. You are allowed to use the internet. (You may use other
references including your notes, calculator, Mathematica, and text.) Read the following paragraph
and sign it.
I have neither given nor received assistance on any of these questions. I understand that the penalty
for cheating on this exam is a D in the course.
Date
Signature
Skills:
1.
x
and y(2) = –3.
2y
Use Euler’s method with a step size of 0.5 to estimate the value of y(4).
Consider the IVP y  
a.
b.
BC 2-3
Solve this IVP and use your solution to determine the exact value of y(4).
Your solution should be of the form where y is a function of x. No TI-89 or
integral tables, please.
Fall12
2. You may use your calculator for this problem, but set-up must be shown clearly. At
time t, a particle moving in the xy-plane is at position  x(t ), y(t )  , where x(t ) and y (t ) are
not explicitly
dx
dy
 4t  1 and
 sin(t 2 ) . At time t = 0, x(0)  0 and y (0)  4 .
dt
dt
(a) Find the speed of the particle at time t = 3, and find the acceleration vector of the
particle at time t = 3.
given. For t  0 ,
(b) Find the slope of the line tangent to the path of the particle at time t = 3.
(c) Find the position of the particle at time t = 3.
(d) Find the total distance traveled by the particle over the time interval 0  t  3 .
BC 2-3
Fall12
3. As Dr. Keyton lectures before 2000 participants at the annual meetings of the
AASMPA, the rate at which the participants fall asleep is proportional to the product of
the # of participants not yet asleep and the square root of the time he has spoken (in
minutes). Suppose there is no one asleep at the beginning of the lecture and after 16
minutes 100 of the participants have fallen asleep.
a. Set up an initial value problem that models the rate at which people fall
asleep. Define all variables.
b. Solve this differential equation.
c. Suppose that Han Lee is in the audience, and that he is neither more likely nor
less likely to fall asleep than any other audience member. Determine the
probability that Han will be awake half way through Dr. Keyton’s two-hour
lecture.
BC 2-3
Fall12
(4)
dy
 2y(1 y) , where
dt
y is the proportion of the population that has heard the rumor at time t.
A certain rumor spreads through a community at the rate of
(a)
What proportion of the population has heard the rumor when it is
spreading the fastest?
(b)
If at time t = 0, ten percent of the people have heard the rumor, find y as a
function of t.
(c)
At what time t is the rumor spreading the fastest?

BC 2-3
Fall12
Concept:
(5)
BC 2-3
dy y  4 x

is not separable, my young masters
dx
x y
y
of mathematics can solve this using the substitution v  . Make it so.
x
Though the differential equation
Fall12