Dr. Sarah's Comments and Peer Review for James and Jeremy's 10
Dr. Sarah's Comments
Change the dfieldplot command to increase the values of y,
sinc there is interesting behavior occuring near 4 and 5.
this is one equilibrium solution, and we can see from the dfieldplot
that this is an unstable solution since other solutions above
and below it tend away from x=.517....
is a stable solution
is an unstable solution
2nd part - good, but let me clarify the y=x curve a bit.
Notice that y=x (as a curve)
is NOT a solution of the de. Let's try plugging
it in to the d.e. to see this: Well, if y=x as a curve
was a solution, then dy/dx=dx/dx =1.
So, the lhs of the de is 1. But the rhs of the de is
y^2-yx = x^2-x^2 when substituting y=x, so we get a 0 on the rhs
of the de. Since 1 is not equal to 0, y=x is not a solution
as a curve of
Let's investigate what happens to a solution with an initial
condition that lies on y=x. (Say y(4)=4, or y(20)=20).
Our initial point (x,x) is on the dfieldplot, and we know that
the solution will not stay along the curve
y=x, so let's see where it goes.
Now we know that initially, the derivative at the point
(x,x) will be 0 via the
rhs of the de. Hence, the solution curve will travel horizontally
a bit to the right (we can see this on the dfieldplot).
Well, once this has occurred, then we are below the y=x
curve and so our solutions x value is now larger than our
solution's y value. It then follows the behavior of the rest
of the solutions that are below the y=x curve - it tends towards
0 because the rhs of the de (y^2 -xy) is negative since x>y.
Then, the solution curve
ends up decreasing.
good but unsure about stability
good job on catching their mistakes. Good job on a very confusing
They seemed to have a very tough problem and did a fairly
good job with it.
worksheet didn't clearly show what was going on
(re-state the problems) but the work looks good and they explained it
1st plot needs larger x and y ranges! Interesting
activity above y=4, nice plots, good description of steps
a few things confusing, but looks good
good presentation although kind of unorganized.
dfieldplot is very useful, good answers to Dr. Sarah's questions
good maple work, good talking
graphs look good, what is the equation at the end?
Didn't understand the explanation of this equation.
What is the DE or where is the DE?
Jumps right into the dfield, good job.
Maple presentation lacks the original diff eqns, so I got a little
confused when it came to the second part.
good, but weren't real sure on their explanation.
They obviously tried hard, but seemed kind of confused on some of the
Worksheet a little confusing - need to related problems and parts somewhere in the
worksheet. It was hard to tell what related to what part. Otherwise,
Hard to follow at start. Need to clarify. Didn't answer all ?'s
You might want to remember to include your de's. Might want to
explain what plots mean. Maple syntax could be cleared up some.
You might explain your syntax a bit more.
They explained their mistakes, it was good!
Slowed down at end and that helped immensely.