Math 282

Quiz 4                                                  Name:                          Solutions                                             

October 15, 2009        

 

1.  Consider the vector field to the right, in which x and y represent the populations of 2 species (in 1000’s).

a)  Explain:  What happens to each of the populations in the short run?  In the long run?

 

In the short run, both populations are decreasing, with y decreasing more rapidly than x.

In the long run, x eventually rebounds and starts to grow again, why y heads to extinction.

(For more detail on x’s growth, see the note below, in part (b) )

 

 

 

 

 

b)  Sketch a graph of x vs. time based on the vector field

(Note:  The graph was intended to show that the solution curve stops at (10,0), in which case x would start at 6, drop to a little below 3, then rise to 10 before leveling off.  Since it was not very clear that the graph did not keep going, I accepted drawings that started as I just described, but did not have x leveling off, but rather had it keep growing).

 

 

 

2.  Use two steps of Euler’s method with step size 0.1 to approximate Y(.2) for the system

x'=x+y, y’=3y,  x(0) = 3, y(0) = 2.  Show all work.

 

t

x

y

x'

y'

0

3

2

5

.1

.5

6

.6

.1

3.5

2.6

6.1

.1

.61

7.8

.78

.2

4.11

3.38

 

 

 

 

 

 

 

3.  Solve the IVP from #2.

The 2nd equation ha solution y=2e3t.  The 1st equation then becomes x’=x+2e3t, or x’-x =2e3t .