Math 3226
Test 2
Fall 1996


No books, notes etc. are permitted.
Show all your work! Box in your answers!
The test has 6 problems and 1 extra-credit problem.

Read the problems very carefully.

1 Consider the following predator-prey model:

eqnarray40

  1. Does x(t) denote the predator population or the prey population?
  2. Find all equilibrium points of the system.
  3. Suppose the prey population becomes extinct while the predator population is still positive. Describe the long-term behavior of the the predator population.
  4. Suppose the predator population becomes extinct while the prey population is still positive. Describe the long-term behavior of the prey population.
  5. Describe the long-term behavior of the system, when the initial populations are given by tex2html_wrap_inline121 and tex2html_wrap_inline123 .

2 Write the second order differential equation

displaymath117

as a system of two first order differential equations.

3 Consider the first order system:

eqnarray54

Is tex2html_wrap_inline125 a solution of the system? Explain!

4 The following depicts the phase portrait of the solution to an autonomous system of differential equations with initial conditions x(0)=2 and y(0)=1. Sketch the time series for x(t) and for y(t) in the coordinate systems provided.

5 Consider the first order system:

eqnarray73

with initial conditions x(0)=-1 and y(0)=2. Use Euler's Method with step size h=0.1 to compute approximations for x(t) and y(t) at time t=0.1 and t=0.2.

6 Consider the first order system:

eqnarray76

  1. Graph the nullclines for this system.
  2. Insert direction field ``arrows" at the nullclines.
  3. Determine the general direction of the direction field in each of the regions cut out by the nullclines.
  4. Sketch possible phase portraits for the two initial conditions indicated by ``dots" for as long as the phase portraits stay within the graph's frame.

  5. Find all equilibrium points of the system.

(Extra Credit) To describe the motion of a falling object close to the earth's surface one normally assumes that air resistance is proportional to the object's velocity.
  1. Model free fall with air resistance under the assumption that air resistance is proportional to the third power of the velocity. (Clearly explain the meaning of all variables and constants you introduce!)
  2. What is the terminal velocity of an object in this model?


Helmut Knaust
Fri Dec 6 12:06:58 MST 1996