BC 3
Quiz #7
Name:
Clearly show ALL appropriate work for full credit.
Calculator allowed.
3 pts
1.
3
Draw possible solution
curves for each of the
given starting values. Be
sure to label each curve.
2
1
a.
A(0, 0)
b.
B (–2, 2)
c.
C (1, –2)
-3
-2
-1
1
-1
-2
-3
Find functions (x) and g(x) so that y = ln x and y = 3x are both solutions to
the differential equation   x   y  g  x   y  6 x .
4 pts
2.
4 pts
3. Solve the IVP:
dy
 y  y  x 2 , y (0)  5
dx
2
3
5 pts
4.
a. The number of supermarkets N(t) throughout the country that are using a computerized checkout
system is described by the logistic model. If in 1980 (take this to be t = 0) there are 1000 stores
using this type of system and there are a total of 20,000 supermarkets in the country, write down
the initial value problem that models the rate at which supermarkets adopt this type of system.
b. Make a sketch of the solution to the above Initial Value problem. Mark the scale on the
vertical axis(# of supermarkets) clearly. You do not need a scale on the time axis.
N
time
9 pts
5. Consider the differential equation
dy
 2x  y .
dx
2
1
1
1
(a) On the axes provided, sketch a slope field for the given differential equation at the six
points indicated.
d2y
(b) Find 2 in terms of x and y. Deteremine the concavity of all solution curves for the
dx
given differential equation in Quadrant II. Give a reason for your answer.
(c) Let y  f ( x) be the particular solution to the differential equation with the initial
condition f (2)  3 . Does f have a relative minimum, a relative maximum, or neither at
x  2 ? Justify your answer.
(d) Find the values of the constants m and b for which y  mx  b is a solution to the
differential equation.

2 pts
6. Find a Maclaurin Series, y   an x n , such that y is a solution to
n 0
dy
 2 xy  0 .
dx