Name
MATH 204 PRACTICE MIDTERM I Please work out each of the given
problems on your own paper. Credit
will be based on the steps that you show towards the final answer.
Show your work. Problem 1 Solve the following
differential equations. A.
(25 points)
x + y B. (25 points) (e^{y} + e^{ x})dx + (e^{y} + 2ye^{x})dy = 0 C.
(25 points)
y
. Problem 2 Currently, Lower Angora
Lake has 300 million gallons of unpolluted water in it.
Due to a gas leak in a motor boat in Upper Angora Lake, there is an
inflow of 5000 gallons per hour that contains 2 grams per gallon of MTBE.
Since it is spring, the outflow of water from Lower Angora Lake is only
4000 gallons per hour. Assume that
the MTBE is completely mixed in the water as soon as it enters the lower lake. A.
Set up a differential equation that describes the rate of change of MTBE
in Lower Angora Lake. B.
Use the result of Problem 1 Part C to determine the amount of MTBE in
Lower Angora Lake one day after the leak began. Problem 3 The Tahoe area has
enough resources to support up to 500 black bears, however if the bear
population declines below 50, the bears will not be able to find mates and will
eventually become extinct. B.
Sketch the direction field for this differential equation and sketch the
solution that corresponds to the initial values:
y(0) = 45, y(0) = 100, and y(0)
= 600.
Problem 4 Please answer the
following true or false. If false,
explain why or provide a counterexample. If
true, explain why. A.
If a differential equation is autonomous then it is separable. B. The initial value problem y'  sin(t^{2}) + e^{t} y = 0, y(1) = 4 is guaranteed to have a unique solution defined for all values of t. C.
The difference equation y_{n+1}
= ky_{n}
has solution y_{n} = y_{0}e^{kn}
D.
d^{3}y
dy
is a third order linear
differential equation. Extra Credit: Write down one thing that your instructor can do to make the class better and one thing that you want to remain the same in the class. (Any constructive remark will be worth full credit.)
