Test 1,Math 3107 Name: _________________________
Spring 2000, Dr. Howard
Please work the problems on the blank paper provided.Staple these sheets
in front of your work when finished.
1.Find the general solution of the differential equation 3y
๎
x4y
๎
๎
2xy.
2.Find the solution of the initial value problem y
๎
x
๎
1
๎
๎
dy
dx
๎
0,y
๎
๎
1
๎
๎
3.
3.Determine which of the following functions is a solution to the differential equation
y
๎๎
๎
4y
๎
0.
y1
๎
e2xy2
๎
sin
๎
2x
๎
y3
๎
cos
๎
4x
๎
4.The figure below depicts the direction field for a particular differential equation. Sketch
a likely curve through the point (0, 4).
-4
-2
0
2
4
t
-4 -2 2 4
x
5.Suppose that an alligator population Pis changing over time at a rate proportional to
the square of P. Given that the swamp contained one dozen alligators in 1988, write an
differential equation with an initial condition modeling the change of the alligator
population with time.
You do not have to solve the differential equation
.
6.Short answer. Cite one of the reasons given in class for which the logistic population
model dP/dt
๎
aP
๎
bP2may be favored to the model for constant relative growth rate:
dP/dt
๎
kP.
7.Consider the differential equation dx
dt
๎
3x
๎
x2.
a.Sketch the associated phase line.
b.For the initial condition x
๎
0
๎
๎
5, determine whether x
๎
t
๎
is increasing, decreasing,
or neither.
c.For the initial condition x
๎
0
๎
๎
5, find lim
t
๎
๎
x
๎
t
๎
.
8.A 120-gallon tank initially contains 90 pounds of salt dissolved in 90 gallons of water.
Brine containing 2 pounds per gallon of salt flows into the tank at the rate of 4 gallons
per minute, and the well-stirred mixture flows out of the tank at the rate of 3 gallons per
minute. How much salt is in the tank after 10 minutes?
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