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Elementary Differential Equations-Math 240 Exam 1-Summer 2007

June 15, 2007

Name……………………… Instructor…………………..

This is a closed book exam. You can use a calculator and 8 ½”X 11” sheet of hand written note (both sides).Detail work must be shown for the full credit.

1. (10 Points). Find all solutions to dydx

xx y y

=+

22 .

2(15 points) Find all solutions to dydx

= − −x yxy

2 2

3, y( )− =1 1 (Solution in implicit form

is accepted)

3(15 Points).Find all solutions to the differential Equation

dydx

y e y yx+ = =13

0 14 , ( )

4. (10 Points) Find the general solution to 2x Siny y e x Cosy y e dydx

x x+ + + =3 2 23 0( ) .

5(10 Points) Solve the following: a) Express (3 – 2i) (4 + 3i) in the form of a bi+ .

b) Express − +−

3 26 8

ii

in the form of a bi+ .

c) Find the modulus of z i= −12

32

.

d) Find the argument of z i= −12

32

.

e) Express z = −12

32

i into polar form

6. (10 Points) Solve the following: a) Write 8 3 in the form of 3 3sin( ) cos( )x + x A xcos( ).ω φ− b) Express 3 2 6cos( )x − π in the form A x bcos( ) sin( ).xω ω+

7(20 Points) The population P t( ) (measured in thousands) of fish is described by the equation

dPdt

P P= − + −2 5 23 2 .P

a) Find the equilibrium point(s) of the equation. Sketch several trajectories indicating carrying capacity and threshold. Classify the equilibrium points as stable or unstable.

b) If the current population of the fish is 2000 how many fish will be there after five years? Explain your reasoning in complete sentences. c) To increase the number of fish it was decided to bring additional 500 fishes to the lake. Is that a wise strategy? Explain. d) The number of fish was decreased to 450 after a fishing season. What is the fate of the population of the fish in the lake? What would you suggest to prevent the extinction? Explain.