BC 3
Quiz #7
Name:
Clearly show ALL appropriate work for full credit.
Calculator allowed.
Find all real values of a such that y  e at is a solution to the differential equation y  5 y  4 y  0 .
4 pts
1.
4 pts
2. Solve the IVP: x 2
dy
 y 2 ( x  1), y (1)  1
dx
9 pts
3.
dy
 2x  y .
dx
On the axes below, sketch a slope field for the given differential equation at the twelve
points indicated, and sketch the solution curve that passes through the point (0,1).
Consider the differential equation
a.
2
1
1
b.
1
2
3
The solution curve that passes through the point (0,1) has a local minimum at x  ln   .
2
What is the y-coordinate of this local minimum?
c.
Let y  f ( x) be a particular solution to this differential equation with the initial condition
f (0)  1 . Use Euler’s Method starting at x  0 with two steps of equal size, to approximate
f (1) . Show steps clearly.
d.
Find
d2y
in terms of x and y. Determine whether the approximation found in part c is less
dx 2
than or greater than the actual value of f (1) . Explain your reasoning.
5 pts
4. Match each slope field with the correct differential equation.
A.
B.
C.
E.
G.
5 pts
D.
F.
H.
1.
y  x  y  2 
2.
y  y  x  2 
3.
y   x  2 y  2
4.
y   y  2 y  2
5.
y   x  y  y  2
6.
y   x  y  y  2
7.
y   y  x  y  2
8.
y   x  y  2  y 
5.
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
2 pts
6. Initially 50 pounds of salt is dissolved in a large tank containing 300 gallons of water. A salt water
solution (brine) is pumped into the tank at a rate of 3 gallons per minute, and then the well stirred
solution is pumped out at the same rate. If the concentration of the solution entering the tank is 2 pounds
per gallon, determine the amount, A(t ) , of salt