Note:  

Solutions are shown in red

 

Below are several problems that have either appeared, or are very similar to problems that have appeared, on previous tests on this material.  This is not a “sample test”, as I did not pay careful attention to the number of problems or how long they could be expected to require, but these questions are of a style and difficulty level comparable to what you can expect to see next week.  As always, the best way to prepare is by doing A LOT of practice problems – use the homework from the book and worksheets as well as the questions below to test your knowledge of the material. 

 

1. a.  Find all equilibria for the system: 

 

4x-2xy=0 →x=0 or y=2

3y-xy=0 →x=3 or y=0

To satisfy both of these conditions, we get (0,0) and (3,2)

 

 

b.  If x and y represent populations of 2 species, what do these equations tell you about the species? (What happens to each population in the absence of the other?  How do they interact?)

 

Each grows exponentially in the absence of the other.  The interactions have a negative impact on each species, so they are competitors.

 

 

 

 

2.  For the system of equations from the previous problem, with initial condition x=4 and y=3:

a. In the short run, what happens to each population? 

b.  Use 2 steps of Euler’s Method with step size t=0.05.

c.  Use your calculator to graph the solution curve in the phase plane.  Interpret this to determine what will happen to each population in the long run (you do not need to draw the curve).

 

a.       If x=4, and y=3,  so both species are dying off initially.

 

b.       

t

x

y

x'

y'

Δt

Δx

Δy

0

4

3

-8

-3

.05

-.4

-.15

.05

3.6

2.85

-6.12

-1.71

.05

-.306

-.0855

.1

3.294

2.7645

 

 

 

 

 

c.       Population x will go extinct, allowing population y to grow exponentially

 

 

3. Solve the partially decoupled system:    

 

 

 

 

4. A damped oscillator is modeled by the differential equation:

a. Rewrite this 2nd order equation as a system of first order equations.

 

b.  Rewrite the system in matrix form.

Y

 

c. What is the characteristic polynomial for this equation?

 

d.  Find a formula for the position of the oscillator as a function of time, given initial conditions of y(0)=1, y'(0) = 0. 

General solution: 

From the initial conditions,

Since the question only asks for the position (not velocity), we can pull off the top row to see that y(t) =

 

5. 

a.  Find the general solution.

 

b. Solve the initial value problem if  

 

6. a.  Find the straight line solutions to the differential equation from the previous problem.

y = x and y = -4x

 

   b.  Graph the phase portrait for this system.  Be sure to show the straight line solutions and typical trajectories.

(I couldn’t draw this with arrows, but you should be able to fill those in!)

 

 

 

7.  The picture below shows the direction field and straight line solutions for the differential equation , but I don't remember what the matrix A is, and the arrows have been left off.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Assuming both eigenvalues are negative,

a.  Is the origin a source, sink, or saddle?  Or do you need more information to answer this?

It is a sink since both eigenvalues are negative.

 

 

b. Use the direction field to help you sketch the phase portrait.

 

 

c.  Suppose that you are told that the eigenvalues for the matrix A are -1 and -6.  Find, with justification, an eigenvector for the eigenvalue -1.

 

Hint:  which straight line are all solutions tangent to as they approach the origin?  What does that tell you in terms of eigenvalues and eigenvectors?