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MATH 204 PRACTICE EXAM 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.

  Printable Key

Problem 1 

Solve the following differential equations.

A.     (25 points)                 x + y
                      y'  =                      
                                  x + 4y

B.     (25 points)    (e^y + e ^-x)dx + (e^y + 2ye^-x)dy  =  0

C.     (25 points)                             y
                      y'  =  A -                      
                                         B - Ct

  .
                                                        A(B - Ct)
            Solution:                    y  =                         + k(B - Ct)^1/C
                                                           1 - C

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. 

A.  Set up an autonomous differential equation that models this situation.

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

Solve the following differential equations.

A.  y'' - 2y' + 2y  =  0,  y(0)  =  0,  y'(0)  =  5

B.  y'' - 4y' + 13y = 0, y(0) = 1, y'(0) = 6

C.  y'' + 12y' + 36y = 0  y(0) = 0, y'(0) = 1

Problem 5

Please answer the following true or false.  If false, explain why or provide a counter-example.  If true, explain why.

A.      If a differential equation is autonomous then it is separable.

B.      The initial value problem y'  =  sin(t²) + 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 = y₀e^kn  

D.            d³y                       dy
     x            + ln(1 - x²)           =  x cos x
           dx³                       dx                          

   is a third order linear differential equation.

E.   The function      y = t sin t

 cannot be a solution of the differential equation.

        y'' + p(t)y' + q(t)y  =  0

with p,and q continuous

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.)